Randy L. Haupt - Wireless Communications Systems An Introduction-Wiley-IEEE Press (2020)

Wireless Communications Systems
Wireless Communications Systems
An Introduction
Randy L. Haupt
Colorado School of Mines
Department of Electrical Engineering
This edition first published 2020
© 2020 John Wiley & Sons, Inc.
Edition History
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Library of Congress Cataloging-in-Publication Data is applied for
9781119419174
Cover Design: Wiley
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10 9 8 7 6 5 4 3 2 1
To the wonderful girls in my life: Sue Ellen, Bonny, Amy, Adeline, and Rose.
You give me love and inspiration
vii
Contents
Preface xiii
Symbols and Acronyms xv
1
1.1
1.2
1.3
1.4
1.5
Introduction 1
Historical Development of Wireless Communications 1
Information 4
Wired Communications 7
Spectrum 9
Communication System 12
Problems 13
References 15
2
Signals and Bits 17
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.13.1
2.13.2
2.14
2.15
2.16
Analog Baseband Signals 17
Digital Baseband Signals 21
Source Coding 22
Line Coding 26
Bandwidth 27
Signal Level 28
Noise and Interference 29
Converting Analog to Digital 36
Channel Coding 39
Repetition 40
Parity Bits 40
Redundancy Checking 42
Error Correcting Codes (ECC) 45
Block Codes 45
Convolutional Codes 47
Interleaving 48
Eye Diagram 50
Intersymbol Interference 51
viii
Contents
2.17
2.18
Raised-Cosine Filter 54
Equalization 57
Problems 62
References 67
3
Passband Signals 71
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.8.1
3.8.2
3.8.3
3.9
3.9.1
3.9.2
3.9.3
3.9.4
Carrier 71
Amplitude-Modulated Signals 72
Frequency-Modulated Signals 80
Phase-Modulated Signals 84
Quadrature Amplitude Modulation 90
Power Spectral Density of Digital Signals 92
BER of Digital Signals 94
Multiplexing in Time and Frequency 94
Frequency Division Multiplexing 95
Time Division Multiplexing 96
Multiple Access 97
Spread Spectrum 100
Interference 101
Frequency-Hopping Spread Spectrum 101
Direct-Sequence Spread Spectrum 103
Code Division Multiple Access (CDMA) 104
Problems 106
References 109
4
Antennas
4.1
4.1.1
4.1.2
4.1.3
4.1.4
4.2
4.2.1
4.2.2
4.2.3
4.2.4
4.3
4.3.1
4.3.1.1
4.3.1.2
4.4
4.5
4.6
111
Signal Properties that Influence Antenna Design
Impedance 111
Gain 112
Polarization 113
Bandwidth 115
Common Antennas 116
Point Sources 116
Wire Antennas 117
Aperture Antennas 125
Microstrip Antennas 128
Antenna Arrays 130
Element Placement 131
Linear Array 131
Arbitrary Array Layouts 134
Electronic Beam Steering 136
Element Pattern 137
Low Sidelobes 138
111
Contents
4.7
4.8
4.9
4.10
4.10.1
4.10.2
4.10.3
4.10.4
4.11
4.11.1
4.11.2
4.11.3
4.11.4
Moving a Null to Reject Interference 140
Null Filling 142
Multiple Beams 144
Antennas for Wireless Applications 146
Handset Antennas 146
Cellular Base Station Antennas 151
Reflector Antennas 156
Antennas for Microwave Links 159
Diversity 162
Spatial Diversity 162
Frequency Diversity 165
Polarization Diversity 165
Time Diversity 166
Problems 166
References 170
5
Propagation in the Channel 173
5.1
5.2
5.3
5.4
5.5
5.6
5.6.1
5.6.2
5.6.3
5.7
5.7.1
5.7.1.1
5.7.1.2
5.7.2
5.7.3
5.7.4
5.8
5.9
5.10
Free Space Propagation 174
Reflection and Refraction 175
Multipath 179
Antennas over the Earth 181
Earth Surface 186
Diffraction 190
Fresnel Diffraction 190
Diffraction from Multiple Obstacles 194
Geometrical Theory of Diffraction 198
Signal Fading 202
Small-Scale Fading Models 205
Rayleigh Fading 205
Rician Fading 209
Approximate Channel Models 212
Large-Scale Fading 214
Channel Ray-Tracing Models 217
Doppler Effects 219
Fade Margin 223
Atmospheric Propagation 224
Problems 234
References 238
6
Satellite Communications
6.1
6.2
6.3
241
Early Development of Satellite Communications 241
Satellite Orbits 245
Satellite Link Budget 254
ix
x
Contents
6.4
6.5
6.6
Bent Pipe Architecture
Multiple Beams 259
Stabilization 261
Problems 262
References 263
7
7.1
7.2
7.3
7.4
7.4.1
7.4.2
7.4.2.1
7.4.2.2
7.5
7.6
7.6.1
7.6.2
7.6.2.1
7.6.2.2
7.6.2.3
RFID 267
259
Historical Development 267
RFID System Overview 270
Tag Data 273
Tag Classes 274
Passive Tags 274
Tags with Batteries or Supercapacitors 277
Semi-Passive Tags 277
Active Tags 278
Data Encoding and Modulation 279
Reader-Tag Communication 281
Near Field 281
Far Field 285
Multiple Readers in an Interrogation Zone 285
Backscatter Communication 288
Chipless Tags 293
Problems 295
References 296
8
Direction Finding 301
8.1
8.1.1
8.1.2
8.1.3
8.2
8.3
8.4
8.5
8.5.1
8.5.2
8.5.3
8.5.4
8.5.5
8.5.6
8.5.7
Direction Finding with a Main Beam 301
Array Output Power 302
Periodogram 304
Wullenweber Array 305
Direction Finding with a Null 307
Adcock Arrays 308
Eigenbeams 310
Direction Finding Algorithms 313
Capon’s Minimum Variance 313
Pisarenko Harmonic Decomposition 315
MUSIC Algorithm 316
Root MUSIC 317
Maximum Entropy Method 318
ESPRIT 319
Estimating and Finding Sources 321
Problems 322
References 322
Contents
9
9.1
9.2
9.3
9.4
9.5
9.6
9.6.1
9.6.2
9.6.3
9.6.4
325
The Need for Adaptive Nulling 325
Beam Cancellation 327
Optimum Weights 328
Least Mean Square (LMS) Algorithm 329
Sample Matrix Inversion Algorithm 332
Adaptive Algorithms Based on Power Minimization 334
Random Search Algorithms 335
Output Power Minimization Algorithms 338
Beam Switching 340
Reconfigurable Antennas 340
Problems 342
References 342
Adaptive Arrays
10
MIMO 345
10.1
10.2
10.3
10.3.1
10.3.2
10.3.3
Types of MIMO 345
The Channel Matrix 349
Recovering the Transmitted Signal Using the Channel Matrix 352
CSIR and CSIT 352
Waterfilling Algorithm 356
CSIR and No CSIT 360
Problems 361
References 362
11
365
Wireless Networks 365
Addresses on a Network 365
Types of Wireless Local Area Networks 367
WLAN Examples 370
Threats 373
Securing Data 376
Cryptography 376
Secret Key Cryptography 379
Public Key Cryptography 379
Hashing 380
Defenses 381
Problems 384
References 385
11.1
11.1.1
11.1.2
11.1.3
11.2
11.3
11.3.1
11.3.2
11.3.3
11.3.4
11.4
Security
12
Biological Effects of RF Fields 389
12.1
12.2
12.3
RF Heating 389
RF Dosimetry 393
RF Radiation Hazards 396
xi
xii
Contents
12.3.1
12.3.2
12.3.3
12.4
12.5
Base Stations 397
Cell Phones 397
Medical Tests 397
Modeling RF Interactions with Humans 398
Harmful Effects of RF Radiation 400
Problems 400
References 401
Appendix A MATLAB Tips 405
A.1
A.2
Introduction 405
Plotting Hint 406
OSI Layers 407
Layer 1: Physical 407
Layer 2: Data Link 407
Layer 3: Network 407
Layer 4: Transport 408
Layer 5: Session 408
Layer 6: Presentation 408
Layer 7: Application 409
Appendix B
B.1
B.2
B.3
B.4
B.5
B.6
B.7
Cellular Generations 411
References 412
Appendix C
Appendix D Bluetooth 413
References 414
Wi-Fi 415
References 416
Appendix E
Software-Defined Radios 419
SDR Basics 419
SDR Hardware 421
SDR Software 422
Cognitive Radio 423
References 423
Appendix F
F.1
F.2
F.3
F.4
Index 425
xiii
Preface
This book targets undergraduate and graduate students as well as professionals
wanting an introduction to wireless communication systems. Wireless systems
pervade all aspects of our lives. I have been an insulin-dependent diabetic most
of my life. Recently, I got a continuous blood glucose monitor (Chapter 11) that
interfaces with my cell phone via Bluetooth (Appendix D). This monitor dramatically changed my life for the better due to the technologies presented in
this book. Even though wireless systems have exponentially expanded over my
lifetime, the future looks even brighter. Learning about wireless systems leads
to a significant advantage over the uninformed.
My career in wireless systems covers a wide range of different projects in
industry, government, and academia. I have taught courses in wireless communications, analog and digital communications, digital signal processing,
probability and statistics, antennas, electromagnetics, software-defined radios,
electronics, electromagnetic compatibility, optimization, and radar at six
different universities and also presented many special topic short courses. I
thought that my diverse background would prepare me for writing a book on
wireless systems. Writing this book humbled me, however. Universities usually
have one or more courses based on each chapter in this book. I learned a lot in
the writing process, and I tried to convey very complex information as clear
and simple manner as possible with plenty of pictures and examples.
This book has two parts: fundamentals (Chapters 1–5) and system applications (Chapters 6–12). The appendices provide some supplemental information
for the reader. I relied on MATLAB for most of the computations and graphics.
There are plenty of examples and pictures illustrating many different aspects
of wireless technology in our lives. Even though I tried to simplify the concepts, students still need some knowledge in electromagnetics, math through
calculus, probability, and linear systems.
The first half of this book delves into theory with many examples of practical applications. Chapter 2 covers data, signals, and digital signal processing.
Chapter 3 introduces analog and digital modulation along with multiplexing
and multiple access techniques. Chapter 4 has an overview of antennas with
xiv
Preface
emphasis on design for wireless systems. The details on antenna array design
are necessary for Chapters 8 and 9 but can be skipped if needed for time constraints. Chapter 5 concerns many aspects of RF propagation from HF to mm
waves.
I selected seven examples of wireless applications for the second half of the
book. These examples make use of the theory introduced in the first half of
the book. Chapter 6 is on satellite communications. My visits to satellite communication facilities and the Smithsonian inspired this chapter. I have worked
on satellite projects over my career. Chapter 7 introduces RFID. This technology
impacts our daily lives and keeps track of many things that we value. Although
I am a novice in this area, I found that a lot of my experience and knowledge in
radar useful in writing this chapter. I have written separate books on the material in Chapters 8 and 9. Here, I provide an overview with practical insights into
the material. Chapter 10 covers the interesting and confusing topic of multiple
input/multiple output (MIMO). I translated this complicated material down to
an understandable level for the nonspecialist. I had fun researching the materials for Chapters 11 and 12. Security and health effects of wireless systems
concern users and designers.
I teach the first five chapters of the book and some of the systems described
in the remaining chapters in my class. I like to have the students do plenty of
MATLAB programming and use hardware experiments to enhance the theory.
My students do final projects and make presentations in lieu of a final exam.
I have students purchase RTL software-defined radios (Appendix F) to have
some hardware to test the theory they learn. My goal is to prepare them for a
job in the wireless technology sector. This goal makes this book unique.
I have extra material available for instructors who adopt this text in the form
of PowerPoint slides, videos, and additional problems. Please contact the publisher to arrange for access.
I owe many thanks to the people who spent time reviewing portions of this
book: Sue Haupt, Payam Nayeri, Bonny Turayev, Amy Shockley, Jake Shockley,
and Mark Leifer. They pushed me to do better.
Randy L. Haupt
Boulder, CO
xv
Symbols and Acronyms
Symbols
A
A
Ae
ALOS
Am
AF
AFN
AFx
AFy
AP
Ap
Ap
AR
a
an
B
b
bn
bn
bpn
BER
C
C
C(𝜈)
̂
C
Cnoise
Cnoise − s
Cs − noise
Cs
low sidelobe taper factor
array steering matrix
effective aperture
amplitude of the LOS signal
amplitude of the message signal
array factor
array factor normalized to N
x-axis array factor
y-axis array factor
antenna pattern
index of geomagnetic activity
physical aperture
axial ration
one dimension of rectangular slot
weights
bandwidth
one dimension of rectangular slot
bit n
weights
parity bit n
bit error rate
channel capacity
covariance matrix
Fresnel cosine integral
sample covariance matrix
noise covariance matrix
noise–signal covariance matrix
signal–noise covariance matrix
signal covariance matrix
xvi
Symbols and Acronyms
Csr
Cst
Ctext
CL
CNR
C/N
c
cp
cdf
cdfnorm
D
D
D
Dmax
Dout
DTE
DTM
d
dh
dia
dmax
dp
drh
dskip
dsp
dth
dx
dy
→
−
E (t)
E
Eantenna
Eb /N 0
Ediff
EGO
Ei
ELOS
Er
Er
Es
ET
Et
Ex
Exco
receive signal covariance matrix
transmit signal covariance matrix
ciphertext
confidence level
carrier-to-noise ratio
carrier-to-noise ratio
speed of light
specific heat
cumulative distribution function
standard normal CDF
antenna directivity
singular value diagonal matrix
raindrop equivalent spherical diameter
maximum aperture size
outer cylindrical conductor of diameter of coax
TE diffraction coefficient for a finitely conducting wedge
TM diffraction coefficient for a finitely conducting wedge
distance between antennas
Hamming distance
wire diameter
maximum distance
penetration depth
distance of receive antenna to horizon
skip distance
spacing between turns in a helical antenna
distance of transmit antenna to horizon
element spacing in x-direction
element spacing in y-direction
time-dependent electric field
error vector
electric field of antenna
energy per bit
diffracted electric field
GO electric field
incident electric field
LOS electric field at receiver
reflected electric field
r component of the electric field
signal energy
total electric field
electric field transmitted into medium
x component of the electric field
co-polarized electric field in the x-direction
Symbols and Acronyms
Excross
Ey
Eyco
Eycross
Ez
E𝜑
E𝜃
EBn (𝜃)
EIRP
ENOB
̂ei
en
̂er
erfc()
F
F(𝜈 F )
F LNA
F pg
F sp
F tag
f
fc
f crit
fD
f Dmax
f hi
f lo
fm
f0
G
G
Gamp
GEGC
GMRC
Gp
Gr
Greader
Gsd
Gt
Gtag
gn
H
H
cross polarized electric field in the x-direction
y component of the electric field
co-polarized electric field in the y-direction
cross polarized electric field in the y-direction
z component of the electric field
𝜑 component of the electric field
𝜃 component of the electric field
eigenbeam n
effective isotropic radiated power
effective number of bits
polarization vector of incident wave
bit error n
polarization vector of receive antenna
complementary error function
noise factor
Fresnel integral
LNA noise factor
path gain factor
frequency spreading factor
fade margin
frequency
carrier frequency
critical frequency
Doppler-shifted frequency
maximum Doppler frequency
highest frequency in a bandwidth
lowest frequency in a bandwidth
frequency of message signal
resonant frequency
generator matrix
antenna gain
amplifier gain
EGC diversity gain
MRC diversity gain
processing gain
gain of receiving antenna
gain of reader antenna in direction of tag
selection diversity gain
gain of transmitting antenna
gain of tag antenna in the direction of reader
generating polynomial
parity check matrix
channel matrix
xvii
xviii
Symbols and Acronyms
Hi
Hn
H
H(f )
H c (f )
H eq (f )
Hn
H o (f )
Hr
Hr
H r (f )
H RC (f )
H RRC (f )
Ht
H t (f )
H𝜑
H𝜃
h
h(t)
′
h
hc (t)
hmn
ho
hr
hRC (t)
hRRC (t)
ht
I
I dipole
IN
I0
I 0 (𝜉)
J
J n (⋅)
KL
K
K
Kp
Kr
k
kB
ke
̂
kmp
incident magnetic field
n × n Hadamard matrix
entropy
transfer function
channel transfer function
equalizer transfer function
screen height
overall transfer function
reflected magnetic field
r component of the magnetic field
receive transfer function
raised cosine transfer function
root-raised cosine transfer function
magnetic field transmitted into medium
transmit transfer function
𝜑 component of the magnetic field
𝜃 component of the magnetic field
height
impulse response
virtual height of the ionospheric layer
channel impulse response function
subchannel impulse response
obstacle height above ground
height of receive antenna
raised cosine impulse response
root-raised cosine impulse response
height of transmit antenna
information
dipole current
N × N identity matrix
constant current
zeroth order modified Bessel function of the first kind
joules
nth order Bessel function
constraint length
kelvin
index of geomagnetic activity
estimated planetary K index
Rice factor
wavenumber
Boltzmann’s constant = 1.38 × 10−23 J/K
earth enlargement constant
unit propagation vector for the multipath signal
Symbols and Acronyms
̂
ktr
kx
ky
kz
k0
L
Lblock
LdB
Ldiff dB
Lfloor
Lhata
Lindoor
Lrain
Lt
LUF
Lwall
l
M
M
MUF
mass
mn
m(t)
N
N0
N(D)
Na
N bits
N cluster
N databits
N dr
Ne
N ec
N ed
N frame
N hel
Nk
N lev
N mess
Np
Nr
Ns
N samp
unit propagation vector from transmitter to receiver
wavenumber in x-direction
wavenumber in y-direction
wavenumber in z-direction
wavenumber in free space
loss
blockage loss
path loss in dB
diffraction loss in dB
floor loss
Hata attenuation
indoor propagation loss
rain loss
transmission line loss
lowest usable frequency
wall loss
number of bits in a message
vector of messages
integer number (quantity)
maximum usable frequency
mass of object
message n
message signal
number of elements
noise power spectral density
rain drop size distribution
number of adaptive element
total number of bits
number of cells in a cluster
number of data bits
Marshall and Palmer drop size constant
electron density
number of errors corrected
number of errors detected
number of frames
number of turns in the helical antenna
rank of channel matrix
number of different quantization levels
number of messages
number of parity bits
number of receive elements
number of samples of the covariance matrix
number of samples
xix
xx
Symbols and Acronyms
N sunfade
Nt
N turn
Nx
Ny
NF
n
n(t)
n
na
ni
nt
P
P
Pa
Pavg
PD
PN
PNamp
PNin
PNout
Pr
Preader
Ps
Psin
Psout
Pt
Ptag
p
pdf
pdfnorm
PI
PN
PNamp
Ps
Psin
Psout
Ptext
PSD
PSDB
PSDP
Q(⋅)
Q
maximum number of sun fade days
number of transmit elements
number of turns in loop antenna
number of elements in x-direction
number of elements in y-direction
noise figure
array noise vector
AWGN with power spectral density N 0 /2
Taylor array constant
atmospheric index of refraction
index of refraction in medium of incident wave
index of refraction in medium of transmitted wave
parity bit generating matrix
power
power absorbed
average power
distortion power
noise power
noise power generated by an amplifier
input noise power
output noise power
power received
reader transmit power
signal power
input signal power
output signal power
transmit power
power delivered to the tag IC
probability
probability density function
standard normal PDF
interference power
noise power
noise power generated by an amplifier
signal power
input signal power
output signal power
plaintext
power spectral density
baseband PSD
bandpass PSD
Q function
eigenvector matrix
Symbols and Acronyms
R
Rain
Rb
Rc
Re
Rf
Ri
Ri
RL
Rload
RLOS
RM
Ro
Rr
r
′
r
ra
rc
re
rec
r𝓁
rmnp
rn
r0
S
S(f )
S(𝜈)
SAR
SB
Sr (f )
St (f )
Swolf
s
s(t)
s[n]
s
sin (t)
sout (t)
sp
sr (t)
sr (t)
̃sr
distance
rainfall rate
data rate
code rate
resistor
co-channel distance
radius of inscribed circle
distance to image antenna
resistive loss
load resistance
LOS path
multipath distance traveled
radius of circumscribed circle
radiation resistance
distance
distance to source point
radius of circular aperture
radius of circular array
radius of the earth
apparent earth radius due to refraction
loop radius
length of path p from transmit element m to receive element
n
radius of the nth Fresnel zone ellipse
distance from center point of diffraction plane
syndrome
Fourier transform of signal
Fresnel sine integral
specific absorption rate
Brussels International Sunspot Number
Fourier transform of received signal
Fourier transform of transmitted signal
Wolf number
separation between wires
analog signal
sampled signal
array signal vector
input signal
output signal
distance from feed to shorting pin on PIFA
receive signal
receive signal vector
recovered data signals at the receiver
xxi
xxii
Symbols and Acronyms
srn (t)
st (t)
st (t)
̃st
̂st (t)
stm (t)
s11
SINAD
SINR
SNR
SNR
SNRin
SNRout
T
T0
T aext
T afeed
T aloss
T ant
Tb
Tc
Te
T line
Ts
T sunfade
t
tan𝛿 LF
U
V
V
V ADCmin
V ADCmax
Vc
VD
V DC
V load
Vm
Vp
V rms
V thresh
vr
vt
w
signal arriving at receive element n
transmit signal
transmit signal vector
transmit data signals before precoding
approximation of transmitted signal
signal transmitted from element m
s-parameter, reflection coefficient
signal-to-noise and distortion ratio
signal-to-interference plus noise ratio
signal-to-noise ratio
average SNR
input SNR
output SNR
period in time
ambient temperature (290 K)
temperature of external sources
temperature of antenna feed line
temperature of antenna losses
antenna temperature
bit length
coherence time
equivalent noise temperature
transmission line temperature
symbol length
maximum sun fade time
time
loss factor
SVD receive data weights
volts
SVD transmit data weights
ADC minimum detectable voltage
ADC maximum allowed voltage
carrier voltage
forward voltage drop
output DC voltage
voltage across load
message signal voltage
peak voltage
rms voltage
threshold voltage
receiver velocity vector
transmitter velocity vector
width
Symbols and Acronyms
w
wn
wopt
X
Xa
Xc
XPD
̂
x
xn
x0
Y
̂
y
yn
y0
Z
Zant
Zc
Zin
ZL0
ZL1
Z0
z
zn
zn
̂z
z0
𝛼
𝛽
𝛽 AM
𝛽 FM
𝛽 PM
Γ
Γd
Γg
Γp (m, n)
ΓTE
ΓTM
Γ0, 1
𝛾
𝛾E
𝛾m
𝛾p
Δ
weight vector
weight at element n
optimized weights
binary information
antenna reactance
decorrelation distance
cross polarization discrimination
unit vector in x-direction
x-location of element n
x-distance in diffraction plane
transmitted codeword
unit vector in y-direction
y-location of element n
y-distance in diffraction plane
received codeword
antenna impedance
characteristic impedance
input impedance
load impedance of binary 0
load impedance of binary 1
free space impedance
z-transform variable
x-location of element n
z-transform of nth zero
unit vector in z-direction
distance from aperture to diffraction pattern
raised cosine filter roll-off factor
constant
amplitude modulation index
frequency modulation index
phase modulation index
reflection coefficient
reflection coefficient from dielectric interface
ground reflection coefficient
reflection coefficient of path p
TE reflection coefficient
TM reflection coefficient
reflection coefficient of binary 0,1
power loss due to distance
Euler’s constant
weight for beam m
packet throughput
time difference
xxiii
xxiv
Symbols and Acronyms
Δf
ΔR
ΔT
Δw[n]
Δ𝜏
𝛿(⋅)
𝛿a
𝛿e
𝛿n
𝛿p
𝜀
𝜀
𝜀(t)
𝜀′r
𝜀′′r
𝜀0
𝜀eff
𝜀r
𝜁
𝜂 ob
𝜂 PCE
𝜂 se
𝜂t
𝜃
𝜃 3dB
𝜃b
𝜃c
𝜃D
𝜃i
𝜃 null1
𝜃r
𝜃s
𝜃t
𝚲noise
𝚲𝜆
𝜆
𝜆Ψ
m
𝜆max
𝜆min
𝜆n
𝜆p
maximum frequency deviation
additional length
rise in temperature
weight increment
additional time
delta function
aperture efficiency
radiation efficiency
phase at element n
polarization loss factor
permittivity
error
error
real part of permittivity
complex part of permittivity
permittivity of free space = 8.854187817 × 10−12 F/m
effective permittivity
relative permittivity
GTD distance parameter
on object gain penalty
RF-DC power conversion efficiency
spectral efficiency
taper efficiency
angle measured from z-axis
3 dB beamwidth
Brewster’s angle
critical angle
diffraction angle
incident angle
location of the first null
reflection angle
scan angle
transmission angle
noise eigenvalues
eigenvalue matrix
wavelength
eigenvalues of 𝚿
maximum eigenvalue
minimum eigenvalue
nth eigenvalue
number of packets
Symbols and Acronyms
𝜆x
𝜆y
𝜆z
𝜆0
𝜆m
𝜇
𝜇0
𝜇
𝜇s
𝜇𝜒dB
𝜈F
𝜈n
𝜉
𝜌
𝜌r
𝜌T
𝜎
𝜎 MP
𝜎 noise
2
𝜎noise
𝜎 rcs
𝜎s
T load
T TE
T TM
𝜏
𝜏 mnp
𝜏n
𝜏p
𝚽s
𝜑
𝜙
𝜙’
𝜙n
𝜙s
𝜒
𝜒 dB
𝚿
𝜓
𝜓g
wavelength in x-direction
wavelength in y-direction
wavelength in z-direction
resonant wavelength
singular value
permeability
permeability of free space = 4𝜋 × 10−7 N/A2
mean
signal mean
mean of 𝜒 dB in dB
Fresnel integral input
noise at element n
integration variable
tissue density
Marshall and Palmer drop size constant
threshold voltage normalized to the rms signal level
standard deviation
standard deviation of multipath signals
noise standard deviation
noise variance
radar cross section of the tag
signal standard deviation
power transmission coefficient
TE transmission coefficient
TM transmission coefficient
time delay
time taken for a signal from transmit element m to receive
element n via path p
time delay at element n
packet length
ESPRIT diagonal matrix
phase
angle measured from x-axis
incidence angle, measured from incidence face
angular location of element n
scan angle
ratio of received to transmitted power
𝜒 in dB
estimate of 𝚽s
phase
angle between ground and signal path
xxv
xxvi
Symbols and Acronyms
𝜓n
𝜓y
𝜓z
Ω
Υ
phase of the nth zero
phase of x-component
phase of y-component
ohms
volume
Acronyms
3DES
A
ABS
ACK
ADC
AES
AF
AFD
AM
AMSAT
AOA
AP
AR
ARQ
ASCII
ASK
AWGN
BAT
BER
BFSK
BPF
bps
BPSK
BSA
BSS
CBC
CDF
CDM
CDMA
CFB
CL
CMOS
CNR
CPS
triple data encryption standard
ampere
acrylonitrile, butadiene, and styrene
acknowledgement
analog to digital convertor
advanced encryption standard
array factor
average fade duration
amplitude modulation
Radio Amateur Satellite Corporation
angle of arrival
access point
axial ratio
automatic repeat request
American Standard Code for Information Interchange
amplitude shift keying
additive white Gaussian noise
battery-assisted tag
bit error rate
binary frequency shift keying
bandpass filter
bits per second
binary phase shift keying
basic service area
basic service set
cipher block chaining
cumulative density function
code division multiplexing
code division multiple access
cipher feedback
confidence level
Complementary metal–oxide–semiconductor
carrier-to-noise ratio
cyber–physical system
Symbols and Acronyms
CRC
CRC
CSI
CSIR
CSIT
CTR
DAC
DARPA
dB
dBm
DBS
DC
DFE
DFE
DL
DMI
DOA
DoS
DPSK
DSB
DSN
DSSS
ECB
ECC
EEPROM
EGC
EHS
EMC
EMI
ENOB
EPC
ESD
ESPRIT
ESS
EUI
FCC
FDD
FDM
FDMA
FDX
FFH
FFT
cyclic redundancy check
cyclic redundancy check
channel state information
channel state information at the receiver
channel state information at the transmitter
counter
digital-to-analog converter
Defense Advanced Research Projects Agency
decibel
power relative to 1 milliwatt
Direct Broadcast Satellite
direct current
direction finding
decision-feedback equalizer
downlink
direct matrix inversion
direction of arrival
denial of service
differential phase shift keying
double sideband
Deep Space Network
direct-sequence spread spectrum
electronic codebook
Error Correcting Codes
electrically erasable programmable read-only memory
equal gain combining
electromagnetic hypersensitivity
electromagnetic compatibility
electromagnetic interference
effective number of bits
electronic product code
electrostatic discharge
Estimation of Signal Parameters via Rotational Invariance
Techniques
extended service set
extended unique identifier
Federal Communications Commission
frequency division duplexing
frequency division multiplexing
frequency division multiple access
full duplex
fast frequency hopping
fast Fourier transform
xxvii
xxviii
Symbols and Acronyms
FM
FM0
FOT
FSS
Gen 2
GEO
GHz
GMSK
GNSS
GO
GPS
GSM
GTD
HDX
HEO
HF
HOW
HPF
HVAC
Hz
IARC
IBSS
IC
IDS
IEEE
IFF
IFFT
IIR
IO
IoT
IP
IPv4
IQ
IR
ISI
ISM
ISO
ITU
IZ
J
JPEG
JPEG-LS
K
frequency modulation
type of encoding
Frequency of Optimum Traffic
Fixed Satellite Service
generation 2
geosynchronous earth orbit
gigahertz
Gaussian minimum shift keying
Global Navigation Satellite System
geometrical optics
Global Positioning System
Global System for Mobile
geometrical theory of diffraction
half duplex
high earth orbit
high frequency
handover
high-pass filter
heating, ventilation, and air conditioning
hertz
International Agency for Research on Cancer
independent basic service set
integrated circuit
intrusion detection system
Institute of Electrical and Electronics Engineers
identify friend or foe
inverse fast Fourier transform
infinite impulse response
input–output
Internet of Things
Internet protocol
Internet protocol version 4
in phase-quadrature
infrared
Intersymbol Interference
industrial, scientific, and medical
International Organization for Standardization
International Telecommunication Union
interrogation zone
joules
Joint Photographic Experts Group
Lossless Joint Photographic Experts Group
kelvin
Symbols and Acronyms
kbps
kg
kHz
LAA
LAN
LBT
LCR
LDS
LED
LEO
LH
LHCP
LMS
LNA
LO
LOS
LPDA
LPF
LRC
LSB
LUF
MAC
Mbps
Mcps
MEM
MEO
MHz
MID
MIMO
MISO
MMSE
MOSFET
MPE
MPEG
MRC
MRI
MSE
MSK
MUF
MUSIC
mW
N
NASA
kilobits per second
kilogram
kilohertz
locally administered address
local area network
listen before talk
level crossing rate
laser direct structuring
light emitting diode
low earth orbit
left hand
left-hand circular polarization
least mean square
low noise amplifier
local oscillator
line of sight
log periodic dipole antenna
low pass filter
longitudinal redundancy check
least significant bit
lowest usable frequency
media access control
megabits per second
million chips per second
maximum entropy method
medium earth orbit
megahertz
molded interconnect devices
multiple input/multiple output
multiple input single output
minimum mean square error
metal-oxide semiconductor field-effect transistor
maximum permissible exposure
Moving Picture Experts Group
maximum ratio combining
magnetic resonance imaging
mean square error
minimum shift keying
maximum usable frequency
MUltiple SIgnal Classification
milliwatt
Newtons
National Aeronautics and Space Administration
xxix
xxx
Symbols and Acronyms
NCAR
NCI
NFC
NIC
NLM
NRZ
NSA
NVIS
OFB
OFDM
OFDMA
OOK
OQPSK
OT
OUI
OWF
PC
PCB
PDF
PHD
PIE
PIFA
PKC
PLF
PM
PRN
PSD
QAM
QPSK
RAM
RAP
RF
RFI
RFID
RHCP
RO
rpm
RTF
RW
RZ
SAR
SBR
SEER
SFH
National Center for Atmospheric Research
National Cancer Institute
near-field communications
network interface card
National Library of Medicine
nonreturn to zero
National Security Agency
near vertical incidence
output feedback
orthogonal frequency division multiplexing
orthogonal frequency division multiple access
on–off keying
offset quadrature phase shift keying
operational technology
organizationally unique identifier
Optimum Working Frequency
personal computer
printed circuit board
probability density function
Pisarenko Harmonic Decomposition
pulse interval encoding
planar inverted F antenna
public key cryptography
polarization loss factor
phase modulation
pseudo random noise
power spectral density
quadrature amplitude modulation
quadrature phase shift keying
random access memory
rogue AP
radio frequency
radio frequency interference
radio frequency identification
right-hand circular polarization
read only
revolutions per minute
reader talk first
read write
return to zero
specific absorption rate
shooting and bouncing rays
Surveillance, Epidemiology, and End Results
slow frequency hopping
Symbols and Acronyms
SFU
SIMO
SIR
SISO
SKC
SMI
SSB
SSID
STRIDE
SVD
TARI
TDD
TDM
TDMA
TDRS
TE
TEC
TEM
TID
TLM
TM
TTF
UAA
UHF
UL
UPC
UTD
UV
UWB
V
VHF
VHP
VPN
VRC
W
WBSAR
WEP
WHO
WLAN
WORM
WPA
WPA2
Solar Flux Units
single input multiple output
signal-to-interference ratio
single-input single-output
secret key cryptography
sample matrix inversion
single sideband
service set identifier
spoofing, tampering, repudiation, information disclosure,
DoS, and elevation of privilege
singular value decomposition
type A reference interval
time division duplexing
time division multiplexing
time division multiple access
Tracking and Data Relay Satellite
transverse electric
total electron count
transverse electromagnetic
tag identifier
telemetry
transverse magnetic
tag talk first
universally administered address
ultra high frequency
uplink
universal product code
uniform theory of diffraction
ultraviolet
ultra wideband
volts
very high frequency
Visible Human Project
virtual private network
vertical redundancy check
watts
whole-body average SAR
Wired Equivalent Privacy
World Health Organization
wireless local area network
write-once-read-many
Wi-Fi Protected Access
Wi-Fi Protected Access version 2
xxxi
1
1
Introduction
At the end of the nineteenth century, “wireless” meant “wireless telegraphy”
which eventually became known as radio. Ham radio kept the term “wireless”
alive, but obscure, until cell phones resurrected it toward the end of the twentieth century. Most wireless technologies use radio frequencies (RF), but infrared
(IR), magnetic, optical, and acoustic systems also enable wireless communication. Wireless systems include a wide range of fixed, mobile, and portable
applications. Designing a wireless system involves all the same challenges as a
wired system plus the antennas and propagation channel. This chapter begins
with a brief history of wireless communications then explains some basic concepts needed for proceeding through the rest of this book. The second half of
this book (Chapters 6–12) is devoted to practical applications.
1.1 Historical Development of Wireless
Communications
Long distance communications seem easy now, but that was not the case
throughout history. In 490 BC, legend says that Philippides ran from Marathon
to Athens and announced that the Greeks defeated the Persians in the
Battle of Marathon (according to Google Maps about a 44.4 km drive), then
dropped dead [1]. That long run became the standard for today’s marathon.
Current wireless networks deliver that same message in a blink of the eye.
People wanted a faster way to communicate over long distances than using a
messenger. Several ingenious, low data rate innovations emerged. Figure 1.1
shows four early wireless communication systems that replaced face-to-face
delivery of the message: smoke signals, heliographs (mirrors), semaphore
(flags), and drums. Weather and limited line of sight hindered most wireless
communications. In addition, messages had to be simple and were prone to
misinterpretation at the receiver.
The first quantum leap in fast long distance communication occurred in the
1800s with the introduction of electrical circuits that send signals over wires. In
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
2
1 Introduction
(a)
(b)
(c)
(d)
Figure 1.1 Early forms of wireless communications. (a) Smoke signals, (b) semaphore, (c)
heliograph, (d) drums.. Source: (a) https://commons.wikimedia.org/wiki/File:Frederic_
Remington_smoke_signal.jpg. Public domain; (b) Author originated; (c) www.photolib.noaa
.gov/historic/c&gs/theb1633.htm. Courtesy of NOAA; (d) https://www.flickr.com/photos/
58034970@N00/178631090. Licensed under CC BY 2.0 [3].
the 1830s, Cooke and Wheatstone demonstrated a telegraph system with five
magnetic needles that an electric current forced to point at letters and numbers
that form a message. Britain adopted this invention for railroad signaling [2]. At
the same time, Samuel Morse independently developed the electric telegraph.
He collaborated with Gale and Vail to build a telegraph that transmitted an electric signal by pushing an operator key that connects a battery to a wire and sends
the electric signal down a wire to a receiver [2]. This simple system required a
switch and battery at both ends of a wire. The length of wire and the loss of the
signal strength over that wire limited communication distance. Wired communications forced users to established nodes, but significantly increased the data
rate as well as made communications independent of weather and line of sight.
Electronic wireless communications began when Maxwell found that all
electromagnetic waves travel at the speed of light. He also discovered the
relationship between electricity and magnetism [4]. Maxwell’s mathematical
ideas of electromagnetic wave propagation needed experimental verification,
so Heinrich Hertz built and tested the 100 MHz dipole antenna shown in
Figure 1.2. In order to increase the radiation intensity in a desired direction,
he built the higher gain reflector antenna shown in Figure 1.3. Hertz provided
1.1 Historical Development of Wireless Communications
Figure 1.2 The first radiating dipole designed by Hertz. Source: https://en.wikipedia.org/
wiki/Heinrich_Hertz#/media/File:Hertz_first_oscillator.png.
Figure 1.3 Hertz designed higher gain reflector
antennas. Source: https://commons.wikimedia.org/
wiki/File:Hertz_spark_gap_transmitter_and_parabolic_
antenna.png.
0
50
100 cm
the means for getting the transmitted signal from a wire to the air then back
to another wire connected to a receiver. Professor Oliver Lodge demonstrated
the reception of wireless Morse code in 1894 using a newly invented “coherer”
or receiver [5]. In 1895, Guglielmo Marconi used a more practical setup to
demonstrate transmitted signals up to one-half mile [6]. Marconi then tried
two new ideas: (i) placing the antenna high off the ground and (ii) grounding
the transmitter and receiver. These modifications demonstrated that signals
could travel up to 3.2 km and over hills. Marconi received a British patent for
radio in 1898 [7] and a US patent a few years later [8]. Around the same time,
Tesla tinkered with radiowave propagation and invented radio remote control
[9]. He transmitted an RF wave from the apparatus shown in Figure 1.4 that
opened and closed switches in order to steer the model boat. Tesla received a
US patent for radio in 1898 [10]. A patent battle between Tesla and Marconi
continued until after their deaths. In 1943 (six years after Marconi’s death and
six months after Tesla’s death), the US Supreme Court ruled that Tesla was the
inventor of radio and not Marconi.
In 1900, Reginald Fessenden demonstrated amplitude-modulation (AM)
radio that allowed more than one station to broadcast at the same time and in
the same area (as opposed to spark-gap radio, where one transmitter covers the
3
4
1 Introduction
Figure 1.4 Tesla’s apparatus for the remote control of
a boat. Source: https://en.wikipedia.org/wiki/Nikola_
Tesla#/media/File:Tesla_boat1.jpg.
entire bandwidth of the spectrum) [11]. A few years later, Edwin Armstrong
patented three important inventions that made today’s radio possible: regeneration, superheterodyning, and wide-band frequency modulation (FM) [12].
Regeneration or the use of positive feedback increased the received radio
signal amplitude to the point where headphones were no longer needed. The
superheterodyne receiver replaced several tuning controls with only one. It
made radios more sensitive and selective as well. Wideband FM improved the
sound quality and fidelity over AM. Armstrong set the stage for the 1940s
when a flurry of inventions made advanced wireless communications possible,
including the mobile phone, spread spectrum, and television. In addition,
Harry Nyquist’s work (Nyquist rate) became the impetus for Claude Shannon
to establish the theoretical foundations for modern information theory [13].
Some of the more notable advances in wireless communications appear in
Figure 1.5.
1.2 Information
A message contains information that a sender wants the recipient to know. The
sender and receiver may be human or not. Some messages are a simple “yes”
or “no,” while others are quite complicated, such as a movie. Message value
depends on the information content. In mathematical terms, the information
content of message n is expressed in bits by [14]
In = log2 (1∕pn ) bits
(1.1)
where pn is the probability of transmitting message n. Thus, a less likely message has a higher information content than a more likely message. The game of
1.2 Information
WW I
Great depression
WW II
First satellite
First man on moon
Internet
Internet of things
1838: Electrical telegraph
1858: First trans-Atlantic telegraph cable
1865: Maxwell’s theory
1880s: Hertz verifies Maxwell’s threory
1876: Telephone (Bell)
1893: Wireless telegraphy
1896: Radio (Marconi)
1898: Remore radio control (Tesla)
1900: First AM voice transmission (Fessenden)
1914: First North American transcontinental telephone calling
1918: Superheterodyne receiver (Armstrong)
1927: Television
1927: First commercial radio-telephone service, U.K.−U.S.
1931: Frequency modulation (RCA)
1933: Birth of radio astronomy (Jansky)
1934: First commercial radio-telephone service, U.S.−Japan
1936: World’s first public videophone network
1941: Spread spectrum (Lamra)
1946: Mobile Telephone
1947: Transistor (Bardeen, Brattain, and Shockley)
1948: “A Mathematical Theory of Communication” (Shannon)
1953: Color television introduced in the US
1956: Transatlantic telephone cable
1960s: US long distance phone network converts to digital
1962: Telstar 1, first commercial communications satellite
1964: Fiber optical relecommunications
1969: Computer networking
1970s: LORAN became the premier radio navigation system
1973: First modern-era mobile (cellular) phone
1979: INMARSAT ship-to-shore satellite communications
1980: 1G
1981: First mobile (cellular) phone network
1990s: Brodcasting converts to digital
1991: 2G
1994: US Army and DARPA started software difined radio
1998: Mobile satellite hand-held phones
1998: 3G
2003: VoIP Internet Telephony
2008: 4G
Soon: 5G
Figure 1.5 Timeline for the development of modern wireless systems.
Scrabble uses this concept to assign points to a letter. In Scrabble, players take
turns placing tiles with letters and points onto a 15 × 15 grid of squares in order
to form words as in a crossword puzzle [15]. Players receive points on the tiles
used to form a word. The letter “Q” has a value of 10, whereas the letter “E” only
has a value of 1, because “Q” occurs less frequently in the English language than
“E.” You know less about a word if it has the letter “E” than if it has the letter “Q.”
Table 1.1 contains the number of letter tiles and associated points in Scrabble.
5
1 Introduction
Table 1.1 Distribution of letters and points in the game of Scrabble [16].
Number of tiles
1
2
0
3
6
8
9
12
O
AI
E
[blank]
1
LSU
2
Points
4
G
3
BCMP
4
FHVWY
5
K
8
JX
10
QZ
NRT
D
The average information called entropy (H) equals the information of message
n times its probability of occurrence summed over all N mess messages.
∑
Nmess
H=
n=1
∑
Nmess
pn In =
pn log2 (1∕pn ) bits
(1.2)
n=1
Example
Calculate the information in the first five letters of the English alphabet given
the graph in Figure 1.6.
14.00%
12.00%
10.00%
8.00%
6.00%
4.00%
2.00%
0.00%
a b c d e f g h i j k l mn o p q r s t u vwx y z
8.167%
1.492%
2.782%
4.253%
12.702%
2.228%
2.015%
6.094%
6.966%
6.966%
0.153%
0.772%
4.025%
2.406%
6.749%
7.507%
1.929%
0.095%
5.987%
6.327%
9.056%
2.758%
0.978%
2.360%
0.150%
1.974%
0.074%
6
Figure 1.6 Frequency of letters in English text [17].
1.3 Wired Communications
Solution
Use (1.1) and the values of pn in Figure 1.6 to generate the following Table 1.2:
Table 1.2 Probability and information associated with the first
five letters of the English alphabets.
Letter
A
B
C
D
E
pn
0.08167
0.01492
0.02782
0.04253
0.12702
In
3.6140
6.0666
5.1677
4.5554
2.9769
1.3 Wired Communications
A transmission line or waveguide minimizes the signal loss by forcing the signal
into a conduit from the transmitter to the receiver. Even wireless systems have
cabling between the transmitter and the antenna or from the antenna to the
receiver.
A single wire only carries a DC current (Figure 1.7a). Time varying signals
need two paths as shown in Figure 1.7b. At one point along the twin wire transmission line, the current on one wire travels in the opposite direction of the
current on the other wire. The fields between the wires add in phase while the
fields outside the wires do not. Thus, the signal stays between the wires as it
propagates from one end to the other. The copper wires have a protective plastic that keeps the wire separation constant. Twisting the two wires, as shown in
Figure 1.7c, reduces coupling from other nearby wires. Cheap two-wire transmission lines work fine at frequencies below 1 GHz. The twin wire transmission
line characteristic impedance is given by [18]
√
( )
𝜇1
s
−1
Ω
(1.3)
cosh
Zc =
𝜀𝜋
dia
where
s = separation between wires
dia = wire diameter
𝜇= permeability
𝜀 = 𝜀r 𝜀0 = permittivity
𝜀r = relative permittivity.
In free space, 𝜇 = 𝜇0 = 4𝜋 × 10−7 N/A2 and 𝜀 = 𝜀0 = 8.854 187 817 × 10−12 F/m.
When all components or lines in an RF circuit have the same impedance, the
maximum power reaches the load. For instance, an antenna that has the same
impedance as the transmission line receives the maximum possible power.
7
8
1 Introduction
(a)
I
−
(b)
I
+
Figure 1.7 Different types of
transmission lines and
waveguides carry signals from
one point to another. (a) Single
wire DC, (b) twin wire, (c) twisted
wire, (d) coaxial cable,
(e) rectangular wave, and
(f ) microstrip.
(c)
(d)
(e)
w
p
tri
os
icr
M
ine
l
Dielectric
substrate
εr
h
(f)
Groundpla
ne
Example
Find the wire separation in a twin wire transmission line with an impedance
of 75 Ω using wires that are 1 mm in diameter and surrounded by plastic with
𝜀r = 2.2.
Solution
Substitute the known quantities into (1.3) and solve 75 =
s = 1.462 mm
377
−1
√ cosh
𝜋 2.2
( )
s
1
⇒
Coaxial cable or coax (Figure 1.7d) appears in many communication systems
above 50 MHz, including satellite and cellular communication systems. Its
advantages include low cost, high bandwidth, and protection from interference. The coax has an inner wire of diameter din surrounded by an outer
cylindrical conductor of diameter Dout . A dielectric surrounding the inner wire
maintains a constant separation between the two conductors. If the dielectric
1.4 Spectrum
Table 1.3 RG-58 coaxial cable attenuation as a function of frequency
(dB/m) [19].
Frequency (MHz)
Loss (dB)
100
500
1000
2500
0.125
0.313
0.478
0.87
is air or gas, then dielectric spacers placed at regular intervals maintain a
constant separation between the conductors. The current on the inner conductor travels in the opposite direction as the current on the inside of the outer
conductor, resulting in the signal propagating in a transverse electromagnetic
(TEM) mode where both the electric and magnetic fields are perpendicular to
the direction of propagation. The coaxial cable has characteristic impedance
given by [19]
√
D
𝜇 1
(1.4)
ln out
Zc =
𝜀 2𝜋
din
Table 1.3 shows the loss in dB/m of RG-58 coaxial cable. The loss increases
with frequency. In contrast, free space loss for wireless systems is independent
of frequency.
A waveguide (like the rectangular metal waveguide in Figure 1.7e) contains
an electromagnetic wave as it propagates from one end to the other. These
reflections form modes that are a function of frequency and the waveguide
dimensions. Among other things, the impedance depends on the shape of the
waveguide and the mode. Optical fibers rely on variations in the dielectric constant of the glass to contain the signal within the fiber.
A PCB (printed circuit board) consists of a thin dielectric sandwiched
between two very thin layers of copper. Microstrip lines have a thin trace
etched from the top layer (Figure 1.7f ). The line width (w), substrate height
(h), and substrate dielectric constant (𝜀r ) determine the line characteristic
impedance [20]. Typically, the impedance is designed to be 50 Ω.
1.4 Spectrum
Signals in wireless communications occupy a designated region of the RF
spectrum. The operating frequency depends on regulatory requirements,
propagation characteristics, signal attenuation, and available bandwidth.
These properties have a distinct impact on key requirements such as the radio
link range and the peak throughput as well as on the system capacity. For
example more bandwidth allows higher throughput. Throughput depends
on the received signal strength. In turn, the received signal strength depends
9
10
1 Introduction
Network/
reach
Visible light
(indoor)
PAN
HAN
WLAN
Beamforming
(outdoor)
Radio
Technologies and range
NFC
RFID
Millimeter wave
Millimeter wave + BF
1 cm
1m
10 m
200 m
WAN Cellular
LPWAN networks
Microwave links
10 km
Figure 1.8 Overview of radio technologies. Source: Burg et al. [21]. Reproduced with
permission of IEEE.
on the propagation characteristics (attenuation) and the maximum transmit
power. On the one hand, higher frequencies have more available bandwidth,
which allows for higher capacity. On the other hand, signal attenuation also
increases proportional to the frequency which limits the range at a given
transmit power. Higher frequencies are generally more attenuated by obstacles
such as walls or windows. Figure 1.8 has an overview of different RF and
technologies with their associated radio range.
A wireless system inserts its signal into the frequency spectrum in order to
reduce interference with the myriad of other users. A party with many people
talking to each other prevents a listener from hearing one particular conversation, because all speakers communicate at baseband. In other words, the frequency components in the signal extend from 0 Hz to some maximum voice
frequency. Converting each conversation to an electrical signal does not help
unless the different signals differ in time, frequency, coding, or polarization.
The message rides on an electrical signal at a higher frequency called a carrier that propagates through the air (wireless) or through a transmission line
or waveguide (wired). Frequency and polarization are properties of the carrier
while time and coding are properties of the information signal. The transmitter
modulates the baseband signal to a higher frequency and the receiver demodulates it.
The radio frequency spectrum extends from about 3 kHz to 300 GHz.
Figure 1.9 shows the playground for various wireless applications. The small
print precludes reading the designated frequency bands but provides an
appreciation for the vast number of applications and the importance of
having sufficient bandwidth to perform the desired function. Go to [22] to
magnify the small print. Fierce and expensive battles occur between users that
want to occupy the same frequency band. Governments auction the rights
(licenses) to transmit signals over specific bands in order to efficiently allocate
the resource as well as to raise money. Currently, commercial applications have
1.4 Spectrum
Figure 1.9 United States frequency allocations in the radio spectrum (courtesy of NTIA) [22].
Source: www.ntia.doc.gov.
needs that conflict with traditional government allocations, such as military
and weather radar bands.
The International Telecommunications Union (ITU) advises national or
regional regulatory bodies that assign and regulate licensed and unlicensed
frequency bands. Licensed bands cover the majority of the spectrum and
require a license for operating wireless systems. While licensing bands are
expensive, the exclusive access avoids uncontrolled interference between users
of the shared medium to provide reliable quality-of-service. Also, regulations
often allow for larger power budgets in licensed bands than in unlicensed
bands since interference is better controlled. In addition to the licensed
spectrum, some frequency bands exist for use by anybody. These unlicensed
parts of the spectrum are known as industrial, scientific, and medical (ISM)
bands. Regulations define a set of rules that enables users to coexist. These
rules typically restrict the maximum amount of transmit power in order to
limit the range of each transmitter and enable spatial reuse of the spectrum.
The severe bandwidth limitations in the microwave spectrum motivate
the use of higher (millimeter wave) frequencies at or beyond 28 GHz.
Technology initially limited use of millimeter wave frequencies, but CMOS
11
12
1 Introduction
(complementary metal–oxide–semiconductor) processing opened consumer
electronics to these frequencies [21]. Another recent push toward millimeter
waves was the worldwide availability of almost 7 GHz of bandwidth at the ISM
band around 60 GHz. Millimeter waves suffer more loss than microwaves, so
they have limited use. Obstacles, including thin walls and windows, highly
attenuate millimeter waves. The 60-GHz ISM band lies close to the oxygen
absorption frequency that induces even more attenuation.
Radiation levels from wireless systems have government specified limits
both in-band and out-of-band. Electromagnetic interference (EMI), also
known as radio frequency interference (RFI), occurs when a device transmits
signals that interfere with another device. Electromagnetic compatibility
(EMC) means that a device does not emit radiation that causes EMI in
other devices. EMI results from conducted and radiated emissions, as well
as electrostatic discharge (ESD). EMC requires all equipment operating in a
common electromagnetic environment to not interfere with each other. Three
approaches to EMC include [23]:
1. Suppress emissions at the source.
2. Make the coupling path as inefficient as possible.
3. Make the receptor less susceptible to the EMI.
Simple solutions, such as grounding and shielding, solve many of these issues.
The Federal Communications Commission (FCC) regulates broadcast
stations, amateur radio operators, and repeater stations in the United States.
In addition, the FCC regulates EMC compliance under Title 47 of the Code
of Federal Regulations [24]. Part 15 of these regulations concerns radio
frequency devices, including intentional transmitters (e.g. mobile phones)
and nonintentional radiators (e.g. PCs and TV receivers). Part 18 concerns
equipment operating in the ISM bands. The FCC requirements only relate to
radiated and conducted emissions. The FCC has no immunity limits like those
associated with European EMC Certification.
1.5 Communication System
This book has two parts. The first part introduces the fundamentals of wireless
communications. The block diagram of the wireless system in Figure 1.10
forms an outline for Chapters 2 through 5. Wireless communication starts with
information. The information might be data or music. An analog-to-digital
converter (ADC) transforms an analog signal into bits. Symbols contain groups
of bits. For example the 8-bit ASCII code for the symbol “1” is 00110001.
Additional bits added to the code detect and/or correct errors. This “channel
coding” allows the receiver to correct errors induced by the channel. The
modulator maps the channel encoder output to an analog signal suitable for
Problems
Analog to
digital
conversion
Coding
Multiplexing/
multiple access
Modulation
Antenna
Analog
information
Digital to
analog
conversion
Channel
Decoding
Chapter 2
Demodulation
Demultiplexing
Chapter 3
Antenna
Chapter 4
Chapter 5
Figure 1.10 Block diagram of a digital wireless communications system.
transmission into the channel. An antenna transmits the signal into the channel
at one end and another antenna receives the signal at the other end. A channel
is the path taken by the transmitted signal to the receiver. Signals become
distorted, noisy, and attenuated in the channel. The demodulator converts the
received analog signal into a digital signal that feeds the channel decoder, etc.
before arriving at the receiver. Successful signal detection occurs when the
signal strength exceeds the receiver threshold and noise and interference did
not induce errors that cannot be corrected.
The second half of this book uses the basic information from the first
half to cover some practical topics in wireless communications. Chapter 6
introduces satellite communications, while Chapter 7 presents radio frequency
identification (RFID). Smart antennas are critical to future advancements in
communications, so Chapters 8–10 cover direction finding, adaptive nulling,
and multiple input multiple output (MIMO). Security (Chapter 11) and
Biological Effects of RF (Chapter 12) are topics of great concern and complete
the book. The appendix has several short topics of interest. Many examples
and problems in this book use MATLAB, so a few MATLAB hints appear in
Appendix A.
Problems
1.1
Calculate the information in the letters A, B, C, D, and E of the English
alphabet using the probabilities in Figure 1.6.
1.2
How many bits do the following pieces of information contain? (a) message probability = 1/2 and (b) message probability = 1.0.
13
14
1 Introduction
1.3
Calculate the entropy of the English alphabet using Figure 1.6.
1.4
Generate a histogram plot for Scrabble that is similar to Figure 1.6,
except the y-axis is (a) points and (b) number of letters.
1.5
Calculate the entropy of Scrabble.
1.6
A meter has a read out of [−5, −3, −1, 0, 1, 3, and 5 V] with corresponding
probabilities of [0.05 0.1 0.1 0.15 0.05 0.25 0.3]. Find the (a) entropy and
(b) entropy if the output is only represented by three levels [−4 V 0 V
4 V].
1.7
Find the entropy of a binary code with two symbols, one with probability
p and the other with probability 1 − p.
1.8
Calculate the entropy of the string “asasdgasdgdsg” based on the frequency of occurrence of the letters in the string.
1.9
A codebook has four messages with probabilities of [0.1 0.2 0.3 0.4]. Find
the number of bits needs to communicate the message using entropy as
the lower bound.
1.10
If four messages have probabilities of [1/8 3/8 3/8 1/8], find the average
information per message.
1.11
If five messages have probabilities of [1/2 1/4 1/16 1/16], find the average
information per message.
1.12
A code uses a dash that is three times as long as a dot and occurs one in
three symbols.
(a) Calculate the information in a dot and a dash.
(b) Calculate the average information of this code.
(c) If a dot lasts 10 ms and the interval between symbols is 10 ms, then
calculate the average rate of information transmission.
1.13
Plot the input impedance of a twin wire transmission line vs. s/din using
(1.3). Assume the wires are enclosed in plastic (find permittivity on web).
1.14
Calculate the input impedance for RG-58 cable. Obtain data for the
calculation from a company on the web. How does your calculation
compare with that given by the company?
References
1.15
Locate an online calculator for microstrip impedance. Find the
impedance of a microstrip line with 𝜀r = 4, h = 0.8, and w = 1.65 mm.
Assume the microstrip trace is 0.035 mm thick.
References
1 The Editors of Encyclopædia Britannica (2015). Battle of marathon. In:
Encyclopædia Britannica, Web.
2 http://www.history.com/topics/inventions/telegrapha (accessed 20 May
2016).
3 https://www.flickr.com/photos/58034970@N00/178631090 (accessed 10
February 2019).
4 Maxwell, J.C. (1873). A Treatise on Electricity and Magnetism. Oxford:
Clarendon Press.
5 https://en.wikipedia.org/wiki/Oliver_Lodge (accessed 25 October 2016).
6 https://en.wikipedia.org/wiki/Guglielmo_Marconi (accessed 25 October
2016).
7 Marconi, G. (1897). Improvements in transmitting electrical impulses and
8
9
10
11
12
13
14
15
16
17
18
signals, and in apparatus therefor. British Patent No. 12,039. Date of Application 2 June 1896; Complete Specification Left, 2 March 1897; Accepted, 2
July 1897.
Marconi, G. (1901). Transmitting electrical impulses and signals and in
apparatus, there-for. US Patent RE11,913, filed 1 April 1901; issued 4 June
1901.
https://en.wikipedia.org/wiki/Nikola_Tesla (accessed 25 October 2016).
Tesla, N. (1900). System of transmission of electrical energy. Issued on,
Patent No. 645,576, entitled 20 March 1900.
https://www.britannica.com/biography/Reginald-Aubrey-Fessenden
(accessed 25 October 2016).
https://en.wikipedia.org/wiki/Edwin_Howard_Armstrong (accessed 25 October 2016).
Shannon, C.E. (1948). A mathematical theory of communication. Bell
System Technical Journal 27, pp. 379–423 and 623–656.
Johnson, D. (2016). Fundamentals of Electrical Engineering I. http://www
.ece.rice.edu/~dhj/courses/elec241/col10040.pdf (accessed 25 May 2016).
https://en.wikipedia.org/wiki/Scrabble (accessed 25 May 2016).
http://www.wordfind.com/scrabble-letter-values (accessed 25 May 2016).
https://en.wikipedia.org/wiki/Letter_frequency (accessed 25 May 2016).
Collin, R.E. (1966). Foundations for Microwave Engineering. New York:
McGraw-Hill.
15
16
1 Introduction
19 http://www.qsl.net/co8tw/Coax_Calculator.htm (accessed 26 May 2016).
20 Pozar, D.M. (1998). Microwave Engineering, 2e. New York: Wiley.
21 Burg, A., Chattopadhyay, A., and Lam, K.Y. (2018). Wireless communication
and security issues for Cyber–Physical Systems and the Internet-of-Things.
Proceedings of the IEEE 106 (1): 38–60.
22 https://www.ntia.doc.gov/files/ntia/publications/january_2016_spectrum_
wall_chart.pdf (accessed 10 December 2018).
23 Paul, C.R. (1992). Introduction to Electromagnetic Compatibility. New York:
Wiley.
24 Ott, H.W. (1988). Noise Reduction Techniques in Electronic Systems, 2e. New
York: Wiley.
17
2
Signals and Bits
Information contains facts about something or someone. It takes the form of
a message sent from one person or machine to another through a communication system. A message contains symbols known to the transmitter and
receiver. For instance, symbols might be letters of the alphabet or words. In
wireless communications, symbols are groups of bits. As long as the transmitter
and receiver understand what the symbols represent, then intelligible communication occurs. If the message becomes corrupted, then the receiver cannot
interpret the correct meaning.
This chapter introduces important concepts about analog signals and their
digital representation. Power levels and bandwidth determine how well the
signal resists noise and interference as well as determine the speed of communication. A communication system tries to successfully transmit as many
messages in the least amount of time. Source coding converts digital baseband
signals into symbols with as few bits as possible in order to increase the information transfer rate in terms of bits per second (bps). Channel coding and
interleaving protect the data bits from noise and interference, so the received
signal has few errors. Sometimes symbols overlap in a message, so steps must
be taken to minimize this intersymbol interference (ISI).
2.1 Analog Baseband Signals
Almost all information starts in a low-frequency analog form at baseband, such
as a voice signal. For example a sensor measures some physical quantity and
outputs an analog signal that corresponds to the information it gathers. Sensors
like a thermometer or microphone represents a low output by a low voltage
while representing a high output by a high voltage. These voltage signals vary
in time like the audio signal in Figure 2.1.
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
2 Signals and Bits
Amplitude (V)
1
0.5
0
−0.5
−1
0
1
2
3
4
5
t (s)
Figure 2.1 Baseband voice signal in the time domain.
A Fourier transform quantifies the frequency content of an analog signal. The
signal in the time domain, s(t), has a Fourier transform given by
∞
S(f ) =
∫−∞
s(t)e−j2𝜋ft dt
(2.1)
A spectrum plots S(f ) vs. f . The complex power spectral density (PSD) with
units of W/Hz has both positive and negative frequencies
PSD = |S(f )|2
(2.2)
Negative frequencies have practical meaning when modulating the signal into a
higher frequency band as presented in Chapter 3. Figure 2.2 is the double-sided
(positive and negative frequencies) PSD for the signal in Figure 2.1. The unit
dBm is dB above one mW.
0
PSD (dBm/Hz)
18
−20
−40
−60
−80
−100
−4
−3
−2
−1
0
f (kHz)
1
2
Figure 2.2 Double-sided PSD of the voice signal in Figure 2.1.
3
4
2.1 Analog Baseband Signals
The one-sided spectrum of s(t) in Figure 2.3 spectrum (f ≥ 0) is found from
the two-sided spectrum by
{
|S(0)|2
f =0
PSD =
(2.3)
|S(f )|2 + |S(−f )|2 f > 0
Integrating the PSD over frequency yields the total power. The inverse Fourier
transform of the spectrum perfectly recovers the time domain signal.
∞
s(t) =
∫−∞
S(f )ej2𝜋ft df
(2.4)
In fact, Parseval’s theorem states that the power in both the time and frequency
representation of a signal are the same:
∞
∫−∞
∞
|s(t)|2 dt =
∫−∞
|S(f )|2 df
(2.5)
Figures 2.2 and 2.3 have the same PSD at 0 Hz. On either side of 0 Hz, however, the complex PSD is half the amplitude of the plot in Figure 2.3. The 1/2
factor come from Euler’s identity, cos(2𝜋ft) = (ej2𝜋ft + e−j2𝜋ft )/2.
The limits of integration of the Fourier and inverse Fourier transforms extend
from −∞ to ∞. In practice, a computer calculates the Fourier transform over
a finite interval. Figure 2.4 is a time-frequency plot of the signal in Figure 2.1
using the MATLAB command
spectrogram(s,kaiser(256,5),220,512,8000,'yaxis')
where
s = signal samples
kaiser = window function
PSD (dBm/Hz)
0
−20
−40
−60
−80
0
0.5
1
1.5
2
2.5
3
f (kHz)
Figure 2.3 Single-sided PSD of the voice signal in Figure 2.1.
3.5
4
19
2 Signals and Bits
4
−30
−50
2
−70
−90
1
PSD (dBm/Hz)
−10
3
f (kHz)
20
−110
0
1
2
3
4
t (s)
Figure 2.4 Time-frequency plot of the voice signal in Figure 2.1.
220 = number of overlapping samples in adjacent Fourier transforms
512 = number of frequency samples in the Fourier transform
8000 = sampling frequency in Hz.
Figure 2.4 results from plotting the Fourier transform of a finite piece (window)
of the time signal in the vertical direction. The window moves in time before
computing and graphing another slice of the Fourier transform. This process
continues until the window reaches the end of the data. Time-frequency plots
reveal how the signal frequency content changes with time.
Example
Show that the power in a 1 ms pulse in the time domain is the same as its power
in the frequency domain.
Solution
Assume that the power is measured across a 1 Ω resistor. In the time domain,
the power is given by
1
∞
∫−∞
|s(t)|2 dt =
∫0
12 dt = 1 W
The Fourier transform of this pulse is
1
S(f ) =
∫0
where sinc(x) =
1e−j2𝜋ft dt =
sin 𝜋x
.
𝜋x
So, the power in the frequency domain is
∞
∫−∞
sin(2𝜋f )
e−j2𝜋f 1 − e−j2𝜋f 0
=
= sinc(2𝜋f )
−j2𝜋f
2𝜋f
∞
|S(f )|2 df =
∫−∞
|sinc(2𝜋f )|2 df = 1 W
The above integral can be solved using MATLAB:
integral(@(f)sinc(f).ˆ2,-1000,1000)
2.2 Digital Baseband Signals
2.2 Digital Baseband Signals
Analog communication keeps the baseband signal in analog form while digital
communication converts the baseband signal to bits via an analog-to-digital
converter (ADC). In either case, modulating the baseband signal to a higher
radio frequency (RF) (passband) prepares it for transmission. The new passband
signal travels through the channel to the receiver that demodulates and decodes
it. Successful wireless communication is possible when the channel capacity
exceeds the source entropy.
Each digital baseband symbol requires a unique combination of bits in order
to distinguish one symbol from another. The sender converts the message into
symbols before transmitting them over the wireless channel. The receiver translates the symbols into the original bits that represent the information being
transmitted. Successful communication requires the transmitter and receiver
know the conversion between symbols and bits.
Figure 2.5 demonstrates the transition from information to message to symbols and finally to a code. A geometrical object has an observable shape. The
English word “circle” describes the observed shape. English symbols (letters)
form a word that represents the shape. A binary string called ASCII (American Standard Code for Information Interchange) code represents each letter.
The resulting binary string has 42 bits and describes the geometric shape called
a circle.
At the transmitter, packet and frame bits sandwich the data bits from the
ADC. The frame functions like the husk that surrounds the hard shell (the
packet) as shown in Figure 2.6. The desirable white coconut meat and milk are
like the data. We want the coconut meat and milk (data) which the shell (packet)
and husk (frame) encapsulate. A packet consists of data bits along with headers
and trailers containing control bits that specify source and destination network
addresses, error detection codes, and sequencing information. An IP (internet
protocol) packet contains control information such as the source and destination IP addresses. A frame surrounds a packet and typically includes frame
synchronization features – a sequence of bits or symbols that indicate to the
Information − object shape
Circle
Message − English word
[c i r c l e]
Symbols − English letters
[1100011 1101001 1110010 1100011 1101100 1100101]
Figure 2.5 Converting information into code.
Code − ASCII
21
22
2 Signals and Bits
Figure 2.6 A coconut resembles
a data frame.
Baseband
signal
Nbits
Frame
trailer
Packet
trailer
ADC
Data
Packet
header
Shell
Coconut
meat and milk
Frame
header
Husk
receiver the beginning and end of the data. A receiver strips the headers from
the frame to get at the packet. Removing the packet headers reveals the data.
2.3 Source Coding
Source coding converts the data bits into a new binary sequence that efficiently
compresses the data before transmission. A source code reduces the redundancy present in the data bits, then represents the data with as few bits as
possible. Codebooks at the transmitter map or encode each possible message
into a unique sequence of bits (Figure 2.7). The receiver decodes the source code
into the original data using the same codebook. Fast communication requires
unambiguously representing a message in as few bits as possible.
Lossless coding (entropy coding) perfectly reconstructs the original data
from the compressed data. For example the entropy code called Lossless
Joint Photographic Experts Group (JPEG-LS) compresses picture and video
signals [1]. Lossy coding approximates data with fewer bits, but the receiver
cannot perfectly reconstruct the original data. It works well for speech, audio,
picture, and video signals that do not need an exact reconstruction. Lossy
source coding techniques typically reduce the number of data bits by an order
of magnitude compared to lossless source coding. Examples of lossy coding
techniques include Joint Photographic Experts Group (JPEG) for still picture
coding [2] and Moving Picture Experts Group (MPEG) for video coding [3].
Message
Code
book
Source
code
Figure 2.7 A codebook converts
between the message and the source
code.
2.3 Source Coding
Reducing the number of data bits in a message causes that message to
transmit faster for a given data rate in bps. If all messages have the same
number of bits (fixed length code), then the number of bits in a message is
independent of its probability of occurrence. Highly efficient source codes
assign short bit sequences to high probability messages and long bit sequences
to low probability messages. Morse code, for instance, assigns one “dot” to the
most common letter “E.” Less common letters have up to four “dashes” and
“dots.”
Example
The basic ASCII code is an example of a fixed length code with 7 bits for each
character. What is the ASCII representation of the message “circle”?
Solution
Use MATLAB to convert the message to ASCII: dec2bin(double
(’circle’))
[1100011 1101001 1110010 1100011 1101100 1100101]
Each letter has a binary string of 7 bits. The six symbols or 42 bits comprise the
message.
An efficient code minimizes the number of bits representing a message. The
function 𝓁(mn ) counts the number of bits in a binary message mn . For instance,
𝓁(10) = 2 and 𝓁(10001) = 5. The expected number of bits in a message (overbar
on 𝓁) is given by [4]
∑
Nmess
𝓁(M) = E[𝓁(M)] =
𝓁(mn )pn
(2.6)
n=1
where the M vector has N mess messages with the probability of the nth message
given by pn . A valid codebook with distinct messages assigned to unique bit
sequences has a lower bound on the number of bits required to encode the
average message [5]:
𝓁(M) ≥ H(M)
(2.7)
where H is the entropy or the average number of bits needed to uniquely communicate the message. Entropy coding assigns fewer bits to more likely messages to get as close to the entropy as possible.
Variable-length codes compress data close to the entropy level with zero error
(lossless data compression). A prefix code is a variable-length code that
[ has no]
other code word for a prefix (initial segment). For example code1 = 1 22 33
is a prefix code[– each code
] word begins with a different number. On the other
hand, code2 = 1 22 122 is not, because “1” is a prefix of “1” and “123,” so they
23
24
2 Signals and Bits
both begin with the same number “1.” A message generated with prefix codes
concatenates code words without any special breaks between symbols.
Example
A receiver gets the message [122].
[
[
]
Is the message unambiguous
for code1 : m1 = 1 m2 = 22 or code2 : m1 = 1
]
m2 = 22 m3 = 122 ?
Solution
code1 : It is unambiguous because this translates to m1 m2
code2 : It is ambiguous because this may translate to either m1 m2 or m3
In 1951, Professor Robert Fano and Claude Shannon were at a dead end in
their research to find the most efficient binary source code [6]. In his class at
MIT, Fano assigned his student David Huffman a final project to develop a new
optimal variable length source code that eluded him and Shannon. The young,
ingenious Huffman envisioned an approach using a frequency-sorted binary
tree. His technique proved to be the most efficient ever reported at the time.
In a single assignment, Huffman bettered his professors (and hopefully got an
“A”). Huffman built the tree from the bottom up instead of the accepted top
down approach, so he did the inverse of the suboptimal codes of the day. Huffman coding uses a specific method for choosing the representation of each
symbol, resulting in a prefix code. The Huffman algorithm has the following
steps [7]:
1. Assume there are N mess possible messages.
2. Sort all possible messages in increasing probability.
m1 , m2 , … , mNmess
p1 ≤ p2 ≤ · · · ≤ pNmess
3. Assign a “1” to the lowest probability message and a “0” to the next lowest
probability message (vise versa is fine).
m1 → 1
and m2 → 0
5. The probability of either of these messages occurring equals the sum of the
probabilities of the two messages.
m1 or m2 → m12 with p12 = p1 + p2
6. Replace m1 or m2 with m12 and p1 + p2 with p12 in the message list.
If more than one message remains, then go to step 2. Otherwise, output the
code words in reverse order.
2.3 Source Coding
Codewords
0
Messages
m5, p5 = 0.5
0
10
110
m4, p4 = 0.25
m3, p3 = 0.125
1
0
1110
m2, p2 = 0.0625
1111
m1, p1 = 0.0625
1
0
1
0
1
p12345
p1234 = 1
2
p123 = 1
4
p12 = 1
8
Figure 2.8 Huffman coding example for five messages.
Example
Assume that five messages have the following probabilities:
p1 = 0.0625,
p2 = 0.0625,
p3 = 0.125,
p4 = 0.25,
p5 = 0.5
Find a Huffman code that represents these five messages.
Solution
Using the Huffman algorithm, assign m1 → 1 and m2 → 0 (or m1 → 0 and
m2 → 1) then combine these messages m1 and m2 → m12 and add their
probabilities p12 = p1 + p2 = 0.0625 + 0.0625 = 0.125. Next, assign m12 → 1
and m3 → 0 then combine these messages m12 and m3 → m123 and add their
probabilities p123 = p12 + p3 = 0.125 + 0.125 = 0.25. Continue the process as
shown in Figure 2.8. Assign a single bit “0” to the highest probability code
word, m5 . The next highest probability code word, m4 , is assigned two bits
“10.” The lowest probability code words have 4 bits each. The receiver knows
that the end of a code word occurs after receiving a zero, “0,” or four ones,
“1111.” Thus, the sequence 0101111 can only have one meaning: m5 , m4 , m1 .
A decoder may confuse code words that closely resemble each other. The
Hamming distance, dh , measures the difference between two strings of the same
length. It counts the number of errors between the transmitted and received
data strings. A binary code also has a Hamming weight which equals the number of nonzero symbols in a code word.
Example
Find the Hamming distance between Symbols 1 and 2 as well as the Hamming
weight of Symbol 1.
25
26
2 Signals and Bits
Solution
Symbol 1
Hamming
distance
Symbol 2
Adeline
Hamming weight of
Symbol 1
Adelphi
3
7
10101010
10111010
1
4
123456789
123546789
2
9
2.4 Line Coding
Line coding, also known as baseband modulation, refers to the voltage signals
that represent bits. There are two categories of line codes: non-return-to-zero
(NRZ) and return-to-zero (RZ). RZ codes have a 0-V level for a portion of every
bit interval, while NRZ codes do not.
Figure 2.9 shows some of the many ways to represent bits by voltage levels
for NRZ and RZ codes. Unipolar NRZ assigns a high voltage level to a “1” and
zero volts to a “0.” This approach is also known as on–off keying (OOK). Polar
1
0
1
1
0
0
1
A
Unipolar NRZ
0
t
A
Bipolar NRZ
0
t
−A
A
Manchester NRZ
0
t
−A
A
Unipolar RZ
0
t
A
Bipolar RZ
0
−A
Figure 2.9 Line coding.
t
2.5 Bandwidth
NRZ represents a one by A volts and a minus one by −A volts. Manchester NRZ
represents a one by A volts in the first half of the bit and −A volts in the second
half. In contrast, a zero is represented by −A volts in the first half of the bit
and A volts in the second half. Unipolar RZ codes represent a one by A volts in
the first half of the bit and 0 V in the second half. A zero is represented by 0 V.
Bipolar RZ represents a zero by 0 V and a one by alternating value of A and −A.
2.5 Bandwidth
Bandwidth (B) defines a frequency range between a low frequency (f lo ) and a
high frequency (f hi ). Sometimes, the bandwidth is expressed as a ratio f hi /f lo ,
while other times, it is expressed as a percent and is called fractional bandwidth:
2(f hi − f lo )/(f hi + f lo ) × 100%. Some set rule defines the high and low frequencies
that bound the bandwidth. In most cases, f lo and f hi occur when the gain (or
some other power specification) decreases by 3 dB on either side of its peak or
center value.
Signal bandwidth contains all significant frequencies in a signal between f lo
and f hi [8]. System bandwidth contains all frequencies between f lo and f hi that
pass through the system without being attenuated below a specified level. In
order to accurately receive the signal, the system bandwidth must be greater
than or equal to the signal bandwidth (Figure 2.10), otherwise some signal frequencies attenuate. In general, the impulse response of the system, h(t), or
transfer function, H(f ), attenuates (or filters) a certain range of frequencies in
the input signal, sin (t), as well as adds noise.
A transmitter sends symbols at a known rate. The receiver detects the symbols then reconstructs the transmitted data. If the symbols have more than 1
bit, then the baud rate (symbols/s) equals the bit rate divided by the number of
bits per symbol. That is if the symbol has 8 bits, the baud rate is one-eighth of
the bit rate. The data rate must be less than twice the bandwidth.
Example
A bit that is T b long has a power spectrum given by Ps = T b sinc2 (fT b ). If f lo and
f hi are defined to be 3 dB below the peak power, then what is the bandwidth of
the signal?
Figure 2.10 System bandwidth
should be greater than signal
bandwidth.
Signal
bandwidth
System
bandwidth
Signal
bandwidth
Sin(t)
h(t)
Sout(t)
Sin(f)
H(f)
Sout(f)
27
28
2 Signals and Bits
Solution
The 3 dB points are found from sinc2 (fT b ) = 0.5 and solving the transcendental
equation for f .
fTb = ±0.443 ⇒ B = 2f = 0.886∕Tb
2.6 Signal Level
At any point in space, the signal level fluctuates over time, making the time
average of the signal amplitude an important property. The mean of an analog
signal, s(t), over a time period, T, is found by integrating the signal over T.
T∕2
1
s(t)dt V
T→∞ T ∫−T∕2
𝜇s = lim
(2.8)
Midpoint integration divides range from −T/2 to T/2 in (2.8) into equal intervals that are Δ long then multiplies the midpoint value by Δ and sums the result.
The mean after sampling s(t) N samp times as shown in Figure 2.11 is
𝜇s =
Nsamp
Nsamp
∑
1
1 ∑
Δs[n] V = lim
s[n] V
Nsamp →∞ ΔN
Nsamp →∞ N
samp n=1
samp n=1
(2.9)
lim
Note that Δ divides out making the result a function of the samples and the
number of samples. Once the signal average is known, then the energy and
power are easy to calculate. Energy in analog form is
T∕2
[s(t) − 𝜇s ]2 dt J
Es = lim
T→∞ ∫−T∕2
(2.10)
and the energy calculated from the sampled signal is
Nsamp
Es =
lim
Nsamp →∞
∑
(s[n] − 𝜇s )2 J
(2.11)
n=1
The power of the analog signal is
T∕2
1
[s(t) − 𝜇s ]2 dt W
T→∞ T ∫−T∕2
Ps = lim
Volts
Figure 2.11 A signal s(t) and its
samples s[n].
s[n]
s(t)
0 1 2 3 4 5...
Δ
(2.12)
t or n
2.7 Noise and Interference
and of the sampled signal is
Ps =
lim
1
Nsamp →∞
Nsamp
Nsamp
∑
(s[n] − 𝜇s )2 W
(2.13)
n=1
The mean, power, and energy are important measurable quantities of a signal.
Example
Assume a “0” is 0 V and a “1” is 1 V and both bits are equally likely. Find the
signal mean, energy, and power with N samp samples.
Solution
𝜇s =
1
Nsamp
Nsamp
∑
s[n] = Nsamp
n=1
1 1 1
= V
Nsamp 2 2
Nsamp ( )
)
∑(
1 2 ∑ 1 2
1
s[n] −
=
= Nsamp = 1 J
2
2
4
n=1
n=1
Nsamp
Es =
Ps =
1
Nsamp
Nsamp ( )
)
∑(
1 2
1 ∑ 1 2
1 1 1
s[n] −
=
= Nsamp
= W
2
N
2
N
4 4
samp
samp
n=1
n=1
Nsamp
2.7 Noise and Interference
Noise and interference make the desired signal difficult to detect. Noise
comes naturally from heat or electron motion. Interference comes from
other signals and may be intentional or unintentional. In either case, the
signal transmitter and receiver try to overcome the noise and interference
through increased power, superior electronic components, coding, and signal
processing. Dependable communication links have a high signal level and a
low noise level. The unitless signal to noise ratio serves as a widely used figure
of merit.
P
(2.14)
SNR = s
PN
where Ps = signal power and PN = noise power. SNR is usually expressed in dB.
SNR = 10 log
Ps
PN
(2.15)
The carrier-to-noise ratio (CNR or C/N) is a version of SNR in which the power
in the carrier is divided by the power in the noise.
29
30
2 Signals and Bits
Thermal noise power arises from vibrations of conduction electrons and
holes and is given by
PN = kB T0 B W
(2.16)
with Boltzmann’s constant k B = 1.38 × 10−23 J/K and T 0 is the temperature
in kelvin (K). Vibrating charges emit spectral content that falls inside the
frequency band of the signals. The thermal noise has a uniform PSD over RF.
Unless otherwise specified, T 0 equals the ambient temperature of 290∘ K.
A resistor at room temperature has an average thermal noise power of
PN = 10 log(1.38 × 10−23 × 290 × 1) = − 204 dBW/Hz = − 174 dBm/Hz.
The noise factor, F, of an amplifier indicates the amount of noise that a device
adds to a signal passing through it and is defined as the ratio of the input SNR
to the output SNR [9].
F=
P ∕P
SNRin
= sin Nin
SNRout
Psout ∕PNout
(2.17)
where
SNRin = input SNR
SNRout = output SNR
Psin = input signal power
Psout = output signal power
PNin = input noise power
PNout = output noise power.
For instance, a perfect amplifier increases the signal and noise by the gain, Gamp ,
so the SNR is the same at the input and output. In the real world, an amplifier
adds noise (PNamp ) and degrades the output SNR. A low F means that a device
does not add much noise.
PNamp + Gamp PNin
Psin ∕PNin
F=
=
Gamp Psin ∕(PNamp + Gamp PNin )
Gamp PNin
PNamp + Gamp kB T0 B
=
(2.18)
Gamp kB T0 B
Note that F is independent of the input signal power. Usually, the noise factor
is expressed in dB as the noise figure (NF).
NF = 10 log F dB
(2.19)
Both PNamp and PNin are proportional to bandwidth, so the bandwidths in all
the terms in (2.18) divide out and make F independent of bandwidth [10].
Since noise power is proportional to the temperature, temperature is an alternative way to characterize the noise. Consider an amplifier having a noise input
power of Pin = kT in B. An imperfect amplifier adds noise characterized by an
2.7 Noise and Interference
equivalent temperature (T e ). As a result, the output noise power of the amplifier is given by
(2.20)
Pout = Gamp k(Tin + Te )B
which means that the output noise temperature is
(2.21)
Tout = Gamp (Tin + Te )
Passive components have an equivalent temperature given by
Te = (L − 1)T0
(2.22)
where L ≥ 0 is the loss with L = 0 being lossless. Amplifier effective noise temperature is
Te = (F − 1)T0
(2.23)
The noise factor which for passive components equals the loss. Noise temperatures are not physical temperatures, but theoretical temperatures that produce
an equivalent noise power. Often times, satellite communications uses a system
temperature to describe the noise performance of a device in place of the NF
(Chapter 6).
The NF of a cascade of amplifiers (Figure 2.12) results from the NFs and
gains/losses of the system components [10]. For one amplifier, the input noise
is k B T 0 B and the output noise is the input noise times the gain plus the noise
added by the amplifier (PN1 ), so the output noise is
(2.24)
PNout = kB T0 BG1 + PN1
Adding another amplifier produces output noise given by
(2.25)
PNout = kB T0 BG1 G2 + PN1 G2 + PN2
G2
T1
...
GNamp
T2
TNamp
PN2
PN1
FNamp
PN1G2
kBT0B
kBT0BG1
kBT0BG1G2
Input
Amplifier 1
Amplifier 2
Figure 2.12 Cascaded amplifiers.
...
PNamp
...
G1
F2
Added noise
F1
PN1G2...GNamp
kBT0BG1G2...GNamp
Amplifier Namp
31
32
2 Signals and Bits
If there are N amp devices, then the output noise is
PNout = kB T0 BG1 G2 · · · GNamp + PN1 G2 · · · GNamp + · · · + PNamp
(2.26)
The output signal power (Psout ) equals the input signal power (Psin ) multiplied
by the gains of all the amplifiers.
Psout = Psin G1 G2 · · · GNamp
(2.27)
Substituting (2.26) and (2.27) into (2.17) yields the equivalent noise factor for
cascaded amplifiers.
FNamp − 1
F −1
Feq = F1 + 2
+···+
(2.28)
G1
G1 G2 · · · GNamp−1
As long as the first amplifier stage in the receiver has a high gain and low noise
factor, then the remaining stages contribute little to the noise that the amplifier
adds to the signal. Placing a low noise amplifier (LNA) first minimizes the noise
induced by the amplifier chain. The equivalent noise temperature for cascaded
amplifiers is given by
TNamp
T
(2.29)
Teq = T1 + 2 + · · · +
G1
G1 G2 · · · GNamp−1
Example
What is the transmission line noise temperature of a coaxial cable with 1 dB of
loss?
Solution
Tline = T0 (1 − 1∕Lt ) = 290(1 − 10−1∕10 ) = 77 K
Additive white Gaussian noise (AWGN) models thermal noise generated
within the communication system. Additive means the noise adds to or
subtracts from the time domain signal. White implies the noise has constant
power across the bandwidth. Gaussian describes the noise power probability
density function (PDF) in the time domain. The PDF for AWGN is given by
PDF(z) =
1
√
𝜎noise 2𝜋
e
−
(z−𝜇N )2
2𝜎 2
N
(2.30)
where 𝜇N is the mean or average and 𝜎 noise is the standard deviation of the
noise. The probability that z lies between a and b equals the integral of the PDF
between a and b.
Example
A square wave signal has a period of 6.25 ns. Plot the signal and signal plus
AWGN having a SNR = 10.
2.7 Noise and Interference
Solution
t = (0:0.1:10)';
x = square(t);
y = awgn(x,10,'measured');
fig(1);plot(t,x,'b-',t,y,'r');xlabel('t (ns)');ylabel
('Amplitude (V)')
legend('Signal','Signal + AWGN');legend('boxoff')
Figure 2.13 is the resulting plot. Electromagnetic interference (EMI) makes
the desired signal more difficult to detect. The signal to interference plus noise
ratio (SINR) measures EMI.
Ps
(2.31)
SINR =
PI + PN
where PI = interference power. Jamming implies intentional interference.
If there is interference but no noise, then the SINR becomes the signal to
interference ratio (SIR). When there is no interference, the SINR becomes
the SNR.
SNR and CNR characterize analog noise. Digital communications systems
frequently use the bit error rate (BER) which equals the number of bit errors
divided by the total number of bits over a long interval. Another common digital quality measure in the baseband signal is Eb /N 0 where Eb is the energy per
bit (W-s) and N 0 is the noise PSD (W/Hz). In other words, Eb /N 0 is the SNR per
bit and serves as a standard of comparison for different modulation methods.
Noise and interference in a digital communication system cause bits to flip
from “1” to “0” or “0” to “1.” The number of bits impacted depends on the data
rate and the time duration of the noise. Errors in a bit stream (Figure 2.14) are
either single bit or multi-bit (burst errors). The multi-bit or burst errors do not
have to be consecutive bits and have a burst length equal to the number of bits
from the first bit error to the last bit error.
Figure 2.13 Signal and
signal plus AWGN having
an SNR = 10.
Amplitude (V)
2
1
0
−1
−2
Signal
Signal + AWGN
0
2
4
6
t (ns)
8
10
33
34
2 Signals and Bits
Single bit error
1
1
0
1
0
0
0
1
1
Transmitted
data
0
1
1
1
0
1
1
0
0
1
1
0
Received
data
1
Burst error
1
0
1
0
0
0
1
1
0
1
1
0
1
0
0
0
1
0
Burst length
Figure 2.14 Single bit vs. burst errors in a bit stream.
Transmitted
X
p(X = 0) 0
p(Y
=
p(X = 1) 1
=
p(Y
Channel
p(Y = 0 | X = 0) = 1−β
1|X
0|X
= 0)
=β
= 1)
=β
Received
Y
0
p(Y = 0) = (1−β)p(X = 0) + βp(X = 1)
p(Y = 1) = (1−β)p(X = 1) + βp(X = 0)
1
p(Y = 1 | X = 1) = 1−β
Figure 2.15 Channel-induced bit errors.
The noise and interference in a channel flips a bit with probability p as shown
in Figure 2.15. The probability that a transmitted “0” becomes a received
“1” or p(Y = 1 ∣ X = 0) equals 𝛽. The probability that a bit stays the same,
p(Y = 0 ∣ X = 0), equals 1 − 𝛽. Thus, the probability of receiving a “1” is given by
Bayes theorem.
p(Y = 1) = p(Y = 1 ∣ X = 1)p(X = 1) + p(Y = 1 ∣ X = 0)p(X = 0)
(2.32)
BER is measured by counting the received bit errors in a long received data
stream. The number of bits in the received data stream needed to accurately
determine the BER depends on the required confidence level (CL) and the
desired BER. The CL equals the percentage of tests where the actual BER is
less than the desired BER. The CL reaches 100% only when there are an infinite
number of bits. The CL is given by [11]
CL = 1 − e−Nbits BER
(2.33)
The number of bits that need to be tested for a given BER and CL is
Nbits =
− ln(1 − CL)
BER
(2.34)
2.7 Noise and Interference
Example
If the specified BER is 10−12 and the required CL is 95%, how many bits need
to be tested?
Solution
Use (2.34) to get
Nbits =
− ln(1 − 0.95)
= 3 × 1012
10−12
In an AWGN channel, a normal (Gaussian) PDF models the probability of the
noise voltage level. The average value of the noise has the highest probability of
occurrence in a normal distribution. The root mean square value of the noise
2
. Typically, the standard normal distribution with
power is the variance or 𝜎noise
a mean of zero and variance of one models noise.
z2
1
PDFnorm (z) = √ e− 2
(2.35)
2𝜋
If the distribution has a mean of 𝜇 and a variance of 𝜎 2 , then its new PDF is
given in terms of the standard normal distribution as
(x − 𝜇)
(x−𝜇)2
1
∕𝜎 = √ e− 2𝜎2
(2.36)
PDF(x) = PDFnorm
𝜎
𝜎 2𝜋
The cumulative distribution function (CDF) results from integrating the PDF.
CDF(xa ) = p(x ≤ xa ) =
xa
∫−∞
∞
1 − CDF(xa ) = p(x ≥ xa ) =
∫xa
2
1
− (x−𝜇)
√ e 2𝜎2 dx
𝜎 2𝜋
2
1
− (x−𝜇)
√ e 2𝜎2 dx
𝜎 2𝜋
and a generic CDF in terms of the unit normal CDF is given by
(x − 𝜇)
CDF(x) = CDFnorm
𝜎
Example
Assume
•
•
•
•
a “0” and a “1” are equally likely
AWGN is zero mean and a variance of 𝜎 2
s(t) < 0.5 the bit is assumed to be “0”
s(t) ≥ 0.5 the bit is assumed to be “1.”
Find the probability of a bit error.
(2.37)
(2.38)
(2.39)
35
36
2 Signals and Bits
0.5 V
0V
II
cdfμ,σ (0.5) = cdf (−0.5 / σ)
1V
I
1-cdfμ,σ (0.5) = cdfμ,σ (−0.5)
= cdf (−0.5 / σ)
Figure 2.16 Gaussian PDFs representing the voltages associated with the received signals.
Solution
The threshold voltage equals the average voltage of 0.5 V. Noise results in normal distributions centered on 0 and 1 V as shown in Figure 2.16. The standard
deviation of the normal distributions define the percent of bits that are misidentified. For instance, the probability that a transmitted “1” is received as a “0”
(Type I error or false positive) equals the integral of the PDF with a mean of 1 V
that lies below 0.5 V:
p(receive 0 ∣ transmit 1) = CDF(0.5) = CDFnorm (−0.5∕𝜎)
Similarly, the probability that a transmitted “0” is received as a “1” (Type II error
or false negative) equals the integral of the PDF with a mean of 1 V that lies
below 0.5 V:
p(receive 1 ∣ transmit 0) = 1 − CDF(0.5) = CDF(−0.5)
= CDFnorm (−0.5∕𝜎)
Thus, the BER is given by
BER = p(error)
= p(receive 0 ∣ transmit 1)p(transmit 1)
+ p(receive 1 ∣ transmit 0)p(transmit 0)
( )
( )
1
1
+ CDFnorm (−0.5∕𝜎)
= CDFnorm (−0.5∕𝜎)
2
2
= CDFnorm (−0.5∕𝜎)
(2.40)
Figure 2.16 shows the Types I and II errors as the shaded areas under the overlapping PDFs.
2.8 Converting Analog to Digital
Converting an analog signal to bits first requires sampling the signal. An ADC
samples the analog signal at a rate of f s which must be at least twice the highest
2.8 Converting Analog to Digital
frequency of the analog signal. The ADC output has N bits bits that define the
quantized voltage levels. The samples fall into one of
Nlev = 2Nbits
(2.41)
quantized voltage levels represented by N bits bits. A 10 bit ADC leads to N lev =
1024. The ADC’s dynamic range extends from the minimum detectable voltage (V ADCmin ) to the maximum allowed voltage (V ADCmax ). ADC input voltage
resolution determines the ADC precision of the analog signal.
VADCres =
VADCmax − VADCmin
Nlev
(2.42)
Figure 2.17 shows the ADC output in bits for input voltages between V ADCmin =
0 to V ADCmax = V max .
The minimum change in voltage corresponds to the least significant bit (LSB)
in the binary signal. A real ADC has noise and distortion that reduce its theoretical resolution. The effective number of bits (ENOB) specifies the ADC
resolution possible in practice with N bits ≥ ENOB.
SINAD − 1.76
dB
(2.43)
6.02
Signal-to-noise and distortion (SINAD) is the ratio of the total power: signal
(Ps ), noise (PN ), and distortion (PD ) to the unwanted power (noise and
distortion).
(
)
Ps + PN + PD
SINAD = 10 log
(2.44)
PN + PD
ENOB =
SINAD indicates the ADC dynamic performance, because it includes all components of noise and distortion.
Vmax
Figure 2.17 The bits assigned to
the quantization levels.
ADC input
0.75 Vmax
0.5 Vmax
0.25 Vmax
0
000 001 010 011 100 101 110 111
ADC output
37
38
2 Signals and Bits
Quantization error is the rounding error between the analog input voltage
and the quantized output voltage corresponding to the bits. The SNR “6 dB
rule” assumes a uniformly distributed quantization error between −1/2 LSB
and +1/2 LSB:
SNR = 20 log(2Nbits ) ≈ 6.02Nbits dB
(2.45)
Applying this equation to a 12 bit ADC leads to a maximum SNR = 72.24 dB.
Example
The Texas Instruments TI 32RF45 data sheet [12] shows that it has f s = 3 GHz,
SNR = 60.9 dB, bandwidth = 3.2 GHz, ENOB = 9.7 bits, SINAD = 60.2 dB, and
V p = 0.625 V. Find its noise figure.
Solution
A full-scale input signal has a peak voltage (V p ) that corresponds to the maximum input voltage of the ADC.
sin (t) = V0 sin(2𝜋ft)
(2.46)
The full-scale power input when the input impedance 50 Ω is given by
Ps =
V2
(0.707V0 )2
= 0
ZADC
100
(2.47)
= 20 log V0 − 20 + 30 = 20 log V0 + 10 dBm
The signal PSD is given by the signal power divided by the bandwidth.
PSD = 20 log V0 + 10 − 10 log(fs ∕2) dBm∕Hz
(2.48)
where f s is the sampling frequency. Assume that the thermal noise at the input
over a 1 Hz bandwidth is −174 dBm/Hz. Usually, the full scale power input is
backed off by 1 dBm, so the input SNR is
SNRin = 20 log V0 + 10 − 1 − 10 log(fs ∕2) + 174
(2.49)
The output SNR is usually specified and is related to the ENOB
SNRout = 1.76 + 6.02 ENOB dB
(2.50)
The noise figure is calculated from (2.49) and (2.50)
NFADC = SNRin − SNRout
= 20 log V0 + 9 − 10 log(fs ∕2) + 174 − 1.76 − 6.02Nbits
= 20 log V0 − 10 log(fs ) − 6.02 ENOB + 181.24 dB
(2.51)
= 24.66 dB
The data Texas Instruments data sheet show that this ADC has a NF = 24.7 dB.
2.9 Channel Coding
2.9 Channel Coding
An RF channel behaves like a water channel. The RF channel takes the RF signal
from the source to its destination. A water channel transfers a well-defined
amount of water proportional to the water channel dimensions. Similarly, an
RF channel transfers an amount of information proportional to its bandwidth.
In reality, the water channel has rocks, turns, and other obstacles that slow the
water and induce turbulence. Similarly, the RF channel has obstacles and noise
that reduce the channel capacity. Claude Shannon’s channel coding theorem
defines a maximum channel capacity as
C = Blog2 (1 + SNR) bps
(2.52)
A channel with noise and interference transmits information at a desired error
level if the channel capacity is not exceeded. Coding techniques get as close as
possible to the Shannon capacity. Spectral efficiency is the data rate (Rb ) in bps
that can be supported by the bandwidth.
𝜂se =
Rb
bps∕Hz
B
(2.53)
Example
If the SNR = 20 dB and the bandwidth available = 4 kHz (voice communications) what is the channel capacity and maximum spectral efficiency?
Solution
C = 4000 log2 (1 + 100) = 4000 log2 (101) = 26.63 kbit∕s
26.63
𝜂se =
= 6.66 bps∕Hz
4
A transmitter starts with N databits data bits in a vector X then transforms them
into a code word Y that has N bits bits with N bits > N databits . Channel coding adds
bits to the data to combat errors. When the code word passes through the channel, the received code word, Z, may have errors.
Z = Y + E = [z1 , … , zNbits ]
(2.54)
where the error vector is E = [e1 , … , eNbits ]. The error vector starts with all zeros.
Position where errors occur changes from “0” to “1.”
The channel coding enables the receiver to either detect errors or detect
and correct errors in the received data bits. A receiver that only detects errors
requests the transmitter to resend a message with an error [13]. Automatic
repeat request (ARQ where the Q stands for query) requires the receiver to
acknowledge receipt of a frame within a timeout period (time period allowed
before an acknowledgment [ACK] is received). Otherwise, the transmitter
39
40
2 Signals and Bits
resends the message. Stop-and-wait ARQ transmits one frame at a time. The
transmitter does not send another frame until it receives an ACK signal is
received by the transmitter. If the ACK does not reach the transmitter before
the timeout, the transmitter resends the same frame. Go-Back-N ARQ sends
N frame frames even without receiving an ACK from the receiver. If the receiver
times out, then the transmitter resends all N frame frames. Selective repeat
transmits N frame frames like Go-Back-N ARQ. In this case, the receiver selectively rejects a single frame, which is retransmitted instead of retransmitting
all N frame frames.
In order to detect and/or correct errors at the receiver, an algorithm converts the N databits into an N bits code word with N bits − N databits parity bits to
detect and/or correct errors. The receiver has a decoding algorithm that recovers the N databits data bits from the code word. The decoding algorithm indicates
if an error occurred and possibly where it occurred in the data. The code rate
for N databits bits encoded into N bits bits is an indication of the efficiency of the
encoding
N
Rc = databits
(2.55)
Nbits
2.10 Repetition
Repetition breaks the data into blocks of bits and transmits them 1/Rc times in
a row. For example the data [1001] repeated three times means the transmitter
sends [100110011001]. If the receiver recovers [100111011001], then it knows
that an error occurred. Repetition efficiency is at most 50% and cannot detect
some errors. For instance, if the receiver recovers [110111011101], which has
the same error in each location, then no error is detected.
Example
What is the code rate for a message of 8 bits repeated five times? Can you think
of an error correction method for the repetition scheme?
Solution
Ndatabits
8
=
= 0.2
Nbits
5×8
One approach to error correction would be majority rule in which the most
common block in the received data is accepted as correct.
Rc =
2.11 Parity Bits
Parity bits tacked onto the data bits detect whether an error occurred in the
transmission. For instance, a code word with 7 data bits (b1 · · ·b7 ) plus 1 parity
2.11 Parity Bits
Data bits
1
1
0
1
0
0
0
1
Odd
parity
A
B
Even
parity
A + B
1
1
0
1
0
1
0
1
1
0
0
0
AB
A + B
Figure 2.18 Generating even and odd parity with XOR gates.
bit (bp1 ) take the form
(2.56)
[b1 b2 b3 b4 b5 b6 b7 bp1 ]
Even parity sets bp1 = 1 when the Hamming weight of the data bits is odd and
bp1 = 0 when the Hamming weight is even. Odd parity sets bp1 = 0 when the
Hamming weight of the data bits is odd and bp1 = 1 when the Hamming weight
is even. One parity bit detects, but does not correct, 1 bit error. For example
if a transmitter sends the data [1011011] with even parity, then bp1 = 1. If the
receiver gets [11111111], it assumes that the data has no errors – a wrong conclusion due to the 2 bit errors in the received data. If the receiver has 1 bit error
[10010111] or 3 bit errors [10010001], for instance, then it correctly assumes
the data is corrupted. A parity detection scheme implemented with XOR gates
is shown in Figure 2.18.
Example
Show the parity bit value for the following data when even and odd parity
is used.
Solution
Parity bit
Data
Even parity
Odd parity
00000000
0
1
01011011
1
0
01010101
0
1
00000001
1
0
01001001
1
0
11111111
0
1
41
42
2 Signals and Bits
2.12 Redundancy Checking
Breaking the data into small blocks then assigning parity bits to the blocks
increases the odds of detecting errors over assigning parity bits to all of the data.
Figure 2.19 shows two approaches to assigning the parity bits [13]: (i) longitudinal redundancy check (LRC) where a parity bit is found for the bits at location
n of each block or (ii) vertical redundancy check (VRC) where a parity bit is
derived for each block. For instance, the 16 bits of data in Figure 2.19 are divided
into four blocks. An LRC even parity bit is found for each of the 4 bit columns
in the blocks. Alternatively, VRC finds an even parity bit for each row in the
blocks. Next, those 4 parity bits append to the original data bits before transmitting (Figure 2.19). On reception, the data separates into blocks then either
the columns (LRC) or rows (VRC) are summed with the parity bits to see if the
result is an even number (for even parity). If not, then errors have corrupted
the data. LRC and VRC increase the likelihood of detecting burst errors. One
error pattern that LRC cannot detect occurs when 2 bit errors in one data block
and 2 bit errors in exactly the same positions in another data block (Figure 2.20).
Some systems use a combination of VRC and LRC for improved error detection.
Data
1
1
0
0
0
0
1
1
1
1
0
0
1
0
1
0
0
0
0
0
1
1
1
1
1
0
1
1
Blocks
VRC
1
0
1
1
1
0
0
1
1
0
1
0
1
0
0
0
1
0
0
1
LRC
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
0
0
Data + LRC
1
0
0
1
1
0
1
0
0
0
1
1
Data + VRC
Figure 2.19 LRC and VRC for a 16 data bits.
Transmitted data
LRC
Received data with errors
1
1
1
1
1
0
0
1
0
1
1
1
1
0
0
1
0
0
1
1
1
0
0
1
0
1
1
1
1
0
1
0
0
0
1
1
1
0
1
1
0
0
1
1
1
0
1
1
Figure 2.20 LRC cannot detect errors when they occur at the same location in two blocks of
the data.
2.12 Redundancy Checking
Example
If the data 11100111110111010011100110101001 has errors at locations 17, 19,
20, 29, 30, and 31, show that LRC will detect the presence of errors.
Solution
Break the transmit and receive data into four blocks as shown in Figure 2.21.
Sum the columns to get the LRC for the transmit data. Do the same for the
receive data. The receive bit sum does not equal the LRC, so the data has errors
and needs to be retransmitted.
Example
Represent the word “running” in ASCII, then use LRC with even parity and
VRC with odd parity to find the parity bits.
Solution
Figure 2.22 shows the results. Note that the ASCII code is a column below the
letter, so the LRC and VRC bits are switched accordingly.
Cyclic redundancy check (CRC) error detection [13] assumes the data bits
serve as coefficients of a polynomial. For instance, the data bits X = [10110] represents x4 + x2 + x. Dividing the polynomial by a binary generating polynomial
Transmit data
Receive data
1
1
1
0
0
1
1
1
1
1
1
0
0
1
1
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
1
0
0
1
1
1
0
0
1
1
0
0
0
1
0
0
1
1
0
1
0
1
0
0
1
1
0
1
0
0
0
1
1
1
0
1
0
1
0
1
0
0
0
0
1
0
0
0
0
LRC
Bit sum
Do not agree
Figure 2.21 Example where LRC finds errors.
Figure 2.22 Using LRC with even parity and VRC
with odd parity for the word “running” encoded in
ASCII.
bit
0
r u n n i n g LRC
0 1 0 0 1 0 1 1
1
1 0 1 1 0 1 1
1
2
0 1 1 1 0 1 1
1
3
0 0 1 1 1 1 0
0
4
1 1 0 0 0 0 0
0
5
1 1 1 1 1 1 1
1
6
1 1 1 1 1 1 1
1
VRC
1 0 0 0 1 0 0
43
44
2 Signals and Bits
(g n ) results in a quotient (discarded) and a remainder (appended to the data as
parity bits to form Y). The receiver recovers the data by dividing the received
bits, Z, by the same generating polynomial. If the remainder is not zero, then an
error occurred in the channel. A zero remainder implies that no error occurred.
CRC detects the following errors [14]:
• single bit errors
• two bit errors as long as the data polynomial has a factor with at least three
terms
• all odd number of errors as long as the polynomial contains the factor x + 1
• all burst errors whose length is less than the length of the remainder
• most large burst errors.
Example
Given the data [100100] and the generator [1101], find the 9 transmitted bits
using CRC.
Solution
The data polynomial is x8 + x5 and the generating polynomial is x3 + x2 + 1.
Since the transmitter sends 9 bits with 6 data bits, the remainder has 3 bits.
Figure 2.23 shows the quotient and remainder from dividing the data polynomial by the generator polynomial. The quotient is discarded and the remainder is appended to the data prior to transmitting. Upon reception, the code
word is divided by the generating polynomial to extract the data as shown in
Figure 2.24. Since the remainder is 0 0 0, the received data has no errors.
Generator
1 1 0 1
1 0 0
1 1 0
1 0
1 1
1
1
1
1
1
0
0
0
1
1
1
Quotient
1 1 1 0 1
0 0 0 0 0
0
1
1
0
1
1
0
0
Codeword
0
1
1
0
1
0
1
1
1 0 0 1 0 0 0 0 1
0
1
1
0
1
1
0
Data
Remainder
0
0
0 0
0 1
0 1
Figure 2.23 Creating the CRC code word for the data polynomial x 8 + x 5 and the generating
polynomial x 3 + x 2 + 1.
2.13 Error Correcting Codes (ECC)
Figure 2.24 Recovering the data
from the CRC code word. A 0 0 0
remainder means there are no errors.
Generator
1 1 0 1
1 0 0
1 1 0
1 0
1 1
1
1
1
1
1
0
0
0
1
1
1
Quotient
1 1 1 0 1
0 0 0 0 1 Codeword
0
1
1
0
1
1
0
0
0
1
1
0
1
0
1
1
0
1
1
0
1
1
0
0
0
0 1
0 1
0 0
2.13 Error Correcting Codes (ECC)
A receiver recovers the original data from an error correcting code (ECC) even
when random errors flip some of the bits. Consequently, the receiver does not
have to ask the transmitter to retransmit data when it detects errors. There are
two main types of ECCs: block codes and convolutional codes.
2.13.1
Block Codes
An (N bits , N dat ) block code maps N dat input bits into N bits output bits as shown
in Figure 2.25, where N bits > N dat [22]. Errors are detected and corrected if the
Hamming distance exceeds N bits + 1 or dh ≥ N ed + N ec + 1, where N ed is the
number of errors detected, and N ec is the number of errors corrected. Since
N ed ≥ N ec , then dh ≥ 2N ec + 1.
A generator matrix, G, encodes binary information with N dat bits, X =
[x1 , … , xNdat ], into an N bits code word, Y = [y1 , … , yNbits ], via
Y = XG
Figure 2.25 A block code has
parity bits for error detection
and correction.
(2.57)
Ndat
Block
coder
Nbits
Data bits
Parity bits
Ndat
Np
Nbits
45
46
2 Signals and Bits
where the generator matrix takes the form
⎡1
⎢0
G=⎢
⎢⋮
⎢
⎣0
0
···
0
g1,Ndat +1
···
1
···
0
g2,Ndat +1
···
⋮
⋱
⋮
⋮
⋱
0
···
1
gNdat ,Ndat +1
···
g1,Nbits ⎤
g2,Nbits ⎥
⎥ = [IN ∣ P]
dat
⋮ ⎥
⎥
gk,Nbits ⎦
(2.58)
G is partitioned into the N dat × N dat identity matrix, INdat , which reproduces the
data at the beginning of the code word, and the N dat × N p parity bit generating matrix, P, which generates parity bits from the data. The generator matrix
creates a systematic code – one that puts X in the first N dat symbols of Y. Each
linear code has a parity check matrix, H, that has the orthogonality properties
YHT = 0
(2.59)
GHT = 0
The parity check matrix decodes the bits when it takes the form
⎡g1,N +1
⎢ dat
H=⎢ ⋮
⎢g
⎣ 1,Nbits
g2,Ndat +1
···
gNdat ,Ndat +1
1
···
⋮
⋱
⋮
⋮
⋱
g2,Nbits
···
gNdat ,Nbits
0
···
0⎤
⎥
⋮⎥ = [PT ∣ INp ]
1⎥⎦
(2.60)
T
H is partitioned by the Np × NNdat transpose of the parity bit matrix, P , and the
N p × N p identity matrix, INp where superscript “T” indicates the transpose of
the matrix.
The receiver decodes Z into a syndrome, S, given by
0
T
GH
S = ZHT = (XG + E)HT = X
+ EHT = EHT
(2.61)
If the syndrome contains all zeros, then there are no errors. The following
example explains how to find the location of the error for a nonzero syndrome.
Example
A block code encodes the data bits X = [1011] with this generating matrix:
⎡1
⎢0
G=⎢
0
⎢
⎣0
0
1
0
0
0
0
1
0
0
0
0
1
0
1
1
1
1
0
1
1
1⎤
1⎥
0⎥
⎥
1⎦
(2.62)
and the receiver has a parity check matrix given by
⎡0
H = ⎢1
⎢
⎣1
0⎤
0⎥
⎥
1⎦
The received data is Z = [1011110]. Find which bit has an error.
1
0
1
1
1
0
1
1
1
1
0
0
0
1
0
(2.63)
2.13 Error Correcting Codes (ECC)
Solution
The encoded information equals the data vector multiplied by the generating
matrix: Y = XG = [1011010]. The syndrome is given by
⎡011⎤
⎢101⎥
⎥
⎢
⎢110⎥
⎥
⎢
S = ZHT = [1011110] ⎢111⎥ = [100]
⎢100⎥
⎥
⎢
⎢010⎥
⎥
⎢
⎣001⎦
This syndrome is nonzero indicating that an error occurred in the transmission. The ST vector is the same as the fifth column in H, which means that
the error occurred in the fifth bit of Y, so E = [0000100]. This approach cannot
correct more than one error. The syndrome for two errors is the sum of the syndromes for the individual errors which incorrectly points to a different column
in H.
The Reed and Solomon systematic code does a linear mapping of a message to
a polynomial [23]. Most two-dimensional bar codes use Reed–Solomon error
correction to recover the bar code even when it is damaged. This approach to
coding is also used in space missions like Voyager [24].
2.13.2
Convolutional Codes
In 1955, Peter Elias introduced nonsystematic codes – codes that have error
detection but do not include the original data [15]. His convolutional code generates a parity bit by performing a mod 2 convolution of the data bits with a
generator polynomial having order K − 1. The polynomial acts like a window
function with constraint length (K). Parity bit i associated with data bit n is
calculated from
(K−1
)
∑
gi [m]x[n − m] mod 2
(2.64)
pi [n] =
m=0
where mod 2 finds the remainder after division of one binary number by
another binary number.
A convolutional code uses generating polynomials to create a code. On average, there are Rc parity bits sent for each data bit. If there are N dat data bits in
a block, then K − 1 previous data blocks must be retained in order to generate
the parity bits. Examples of generator polynomials for rate 1/2 convolutional
codes with different constraint lengths are shown in Table 2.1.
47
48
2 Signals and Bits
Table 2.1 Generator polynomials for
R = 1/2 convolutional codes [16].
K
g1
g2
3
110
111
4
1101
1110
5
11010
11101
6
110101
111011
7
110101
110101
8
110111
1110011
9
110111
111001101
10
110111001
1110011001
Example
A (N bits , N dat , K) = (2, 1, 2) convolutional code has the generating polynomials g 1 = [111] and g 2 = [101]. Find the output code word given the input data
X = [11101].
Solution
First, convolve g 1 with X: g 1 * X = [111] * [11101] = [1010011].
Next, convolve g 1 with X: g 2 * X = [101] * [11101] = [1101001].
Finally, interleave the bits from the two convolutions to get the code word:
[
]
Y = 11 01 10 01 00 10 11
The code rate is Rc = 5/14 = 0.357.
In order to decode the code word, the receiver compares the received bits to
a list of possible bit sequences. Decoders, such as the Viterbi algorithm [17],
determine the most likely bit sequence that was transmitted. A longer code has
better error-correcting performance at the cost of a more complex decoding
algorithm and decreased data rate. In the early 1990s, turbo codes revolutionized forward error corrections (FECs) by approaching the Shannon limit. The
Cassini Saturn probe, Mars Pathfinder, and Mars Rover used convolutional
coding with Rc = 1/6 and K = 15 and Viterbi decoding [19]. Turbo codes found
their first practical application in the Mars Pathfinder mission [21].
2.14 Interleaving
Interleaving mixes up symbols from code words in the same bit stream in order
to convert burst errors into single bit errors are detected and corrected using
ECC [21]. Consider the case of four code words having four symbols each.
2.14 Interleaving
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16
(a) Data before interleaving
1
5
9
13
2
6
10 14
3
7
11 15
4
8
12 16
(b) Two-dimensional 4x4 array used for interleaving
1
5
9
13
2
6
10 14
3
7
11 15
4
8
12 16
(c) Data after interleaving
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16
(d) Data after de-interleaving
Figure 2.26 Example of 1-D block interleaving. Source: Reprinted by permission of [21];
© 2004 IEEE [17].
The 16 consecutive symbols appear in Figure 2.26a. Block interleaving, first
creates a 4 × 4 2-D array in which each code word occupies a column of the
matrix shown in Figure 2.26b. The interleaved code symbols result from writing the code symbols row-by-row from the matrix in Figure 2.26b. The result
appears in Figure 2.26c. Assume an error burst affects four consecutive symbols
(shaded in Figure 2.26). De-interleaving effectively spreads the burst error over
four code words, resulting in one error in each of the four code words as shown
in Figure 2.26d. Hence, the burst error appears as random single bit errors.
Example
Create a short ASCII message in MATLAB and use (12,7) cyclic encoding. Corrupt the message with a burst error. Compare the received bits with and without
interleaving.
Solution
1. Create message:
rndsd=1234; % random seed
word='wireless';
49
50
2 Signals and Bits
a=dec2bin(double(word)); % convert to decimal
numsym=length(word); % number of symbols in message
b=reshape(a,numsym*7,1); % binary message in a
%column vector
msg=de2bi(str2num(b)); % message is in binary
2. Use cyclic coding:
n = 12; k = 7; % cyclic code rate
cpoly=cyclpoly(n, k); % create generating polynomial
code = encode(msg,n,k,'cyclic',cpoly); % Encoded data
3. Create a 6-bit burst error:
errors = zeros(length(code),1); errors(n-2:n+2) =
[1 1 1 1 1]';
4. Interleaving:
inter = randintrlv(code,rndsd); % Interleave.
intererr = bitxor(inter,errors); % Include burst error.
deinter = randdeintrlv(intererr,rndsd); % Deinterleave.
decoded = decode(deinter,n,k,'cyclic',cpoly); % Decode.
disp('Interleaving:');
[number,BER] = biterr(msg,decoded) % Error statistics
5. No interleaving:
code_err = bitxor(code,errors); % Include burst error.
decoded = decode(code_err,n,k,'cyclic',cpoly); % Decode.
disp('No interleaving');
[number,BER] = biterr(msg,decoded) % Error statistics
Interleaving has zero errors and a BER = 0.0.
No interleaving has four errors and a BER = 0.0714
Results depend upon the value of rndsd.
2.15 Eye Diagram
An eye diagram consists of overlaying oscilloscope sweeps from consecutive
segments of a long data stream. Overlaying positive and negative pulses results
in an image that looks like an eye. Ideally, eye diagrams look like rectangular
boxes. In the real world, filters round the sharp edges of rectangular pulses.
Differences in timing and amplitude from bit to bit cause the eye opening to
shrink. Figure 2.27 has two examples of measured eye diagrams. Figure 2.27a
has a large eye opening that corresponds to a large SNR. Reducing the SNR, as
in eye diagram in Figure 2.27b, causes the eye to close. Optimal sampling occurs
2.16 Intersymbol Interference
Tb
Figure 2.27 Eye diagrams.
(a) Low SNR and jitter and
(b) higher SNR and jitter.
−0.50 sym, +0.76
SNR
Eye
Jitter Optimal
sample time
(a)
(b)
in the center of the eye. The start of each bit varies, so the bits do not appear
to have a common starting point. This timing variation from bit to bit is called
jitter. The eye diagram in Figure 2.27b has more jitter than the eye diagram
in Figure 2.27a. The hexagon that roughly outlines the inside of the eye has a
vertical opening proportional to the SNR and a horizontal opening relating to
the sensitivity of the sampling time. A large eye opening corresponds to a low
BER. The steeper the slope of the hexagon sides, the more timing jitter it has.
2.16 Intersymbol Interference
Multipath, noise, and interference distort symbols passing through the channel.
The channel does not treat all frequency components the same (i.e. the channel is dispersive). A dispersive channel has an impulse response and noise that
changes the transmitted signal by the time it arrives at the receiver.
sr (t) = st (t) ∗ hc (t) + n(t)
(2.65)
where
sr (t) = received signal
st (t) = transmitted signal
hc (t) = channel impulse response function
n(t) = AWGN with PSD N 0 /2.
A bandlimited channel behaves like a low pass filter (LPF) that smears the
transmitted signal in time so that adjacent symbols overlap. The spreading of
one symbol into an adjacent symbol causes ISI that leads to wrong symbol
51
52
2 Signals and Bits
identification. Delay spread is the difference between the arrival time of the
line-of-sight (LOS) signal and the arrival time of the last multipath signal. ISI
can be ignored when the symbol duration is much larger than the delay spread.
A signal takes multiple paths of different lengths to the receiver. A longer path
usually has more attenuation than a shorter path due to more reflections. The
sum of all the multipath signals at the receiver has a long tail compared to the
transmitted signal. This tail extends into the adjacent bit causing ISI.
Figure 2.28a displays a sequence of 8 bits originating at the transmitter. Each
bit is T b = 1 μs long. These bits pass through a LPF that rounds the sharp edges
and spreads the symbols in time. If there is no multipath, the individual pulses
arrive at the receiver as shown in Figure 2.28b. The receiver samples the pulses
at the center of the pulse or every 1 μs. These orthogonal pulses have no overlap
at the sample points, so no ISI occurs when the pulses add together as shown
in Figure 2.28c. In this case, 1 V represents a “1” while 0 V represents a “0.”
Multipath in the channel leads to increasing the pulse duration due to multiple
versions of the pulses arriving at different times at the receiver. Figure 2.28d
shows the individual received dispersed pulses. Dispersion ruined the pulse
orthogonality, so that when one pulse is a “1” the rest are no longer a “0.” Adding
the dispersed pulses together produces the waveform in Figure 2.28e. The ISI
produces 0.4 V at the 2 μs sampling point which is a “0.” Now, the sample is
about 0.4 V instead of 0 V. Adding noise to this scenario might well push this
sample point above 0.5 V, so that it is interpreted as a “1” by the receiver.
Figure 2.28 demonstrates that ISI potentially leads to bit errors. In order to
reduce bit errors due to ISI, system designers use the following remedies:
1. Slow down the data rate until the symbol duration is much larger than the
delay spread.
2. Use ECC.
3. Use pulse shaping of the symbols in order to minimize the spread into adjacent symbols.
4. Apply an equalizer that reverses the distortion caused by the channel.
Slowing the data rate is usually not an acceptable solution due to the demand for
higher data rates. Since ECC has already been presented earlier in the chapter,
pulse shaping and the equalizer will be discussed in Section 2.13.
Figure 2.29 diagrams the transmit (H t ), channel (H c ), and receive (H r ) filter
transfer functions in a dispersive channel [25]. Let the overall transfer function
be represented by
Ho (f ) = Ht (f )Hc (f )Hr (f )
(2.66)
Minimizing ISI requires a constant H o (f ) over all frequencies. A constant
H o (f ) results from designing H t and H r for a given H c . A matched receive filter
has the transfer function:
Hr (f ) = Ht∗ (f )Hc∗ (f )
(2.67)
2.16 Intersymbol Interference
0
0
0
1
1
1
0
1
0
−4
−3
−2
−1
0
t (μs)
1
2
3
(a)
(b)
Pulses (V)
1
0.5
0
(c)
Pulses (V)
1
0.8
0.6
0.4
0.2
0
(d)
Pulses (V)
1
0.5
0
(e)
Pulses (V)
1.5
1
0.5
4
Figure 2.28 ISI example for a digital waveform. (a) Ideal pulses for each bit, (b) individual
filtered pulses with no multipath, (c) sum of the received raised-cosine pulses with no
multipath, (d) individual received raised-cosine pulses with multipath, and (e) sum of the
received raised-cosine pulses with multipath.
53
54
2 Signals and Bits
Ho(f)
St(f)
Transmit
Channel
Ht(f)
Hc(f)
Figure 2.29 Dispersive channel
model.
Receive
+
Hr(f)
Sr(f)
AWGN
where “*” indicates complex conjugate. Selecting H t (f ) to make H r (f ) constant
requires knowledge of the channel transfer function. Three approaches to making H r (f ) constant are
1. A pulse-shaping filter in the transmitter.
2. Pick H t (f ) and design H r (f ) to make H o (f ) constant.
3. Pick a simple filter for H r (f ) then digitize the output. The digital output goes
to an equalizer or digital filter that compensates the frequency response. In
this way, the digital filter adapts to a changing channel.
Correlation measures the similarity of two things. A high correlation
implies that two things are very similar. Correlating the received signal with a
matched filter is the same as convolving the received signal with a conjugated
time-reversed version of itself. When the matched filter and received signal
align, the output of the correlator/matched filter peaks. A matched filter
[25, 26] maximizes the receiver SNR in the presence of AWGN. It has an
impulse response that is a time reversed complex conjugate of the transmitted
signal [27].
2.17 Raised-Cosine Filter
A raised-cosine filter, also known as the Hann window in digital signal processing, shapes a digital pulse at the transmitter in order to reduce ISI. It has a
frequency response given by [22]
⎧1
⎪ {
)]}
[
(
𝜋Ts
⎪1
1−𝛼
HRC (f ) = ⎨
1 + cos
|f | −
𝛼
2Ts
⎪2
⎪0
⎩
×0≤𝛼 ≤1
|f | ≤
1−𝛼
2Ts
1+𝛼
1−𝛼
< |f | ≤
2Ts
2Ts
1+𝛼
|f | >
2Ts
(2.68)
where 𝛼 is the roll-off factor that determines the bandwidth of the spectrum.
This filter has a flat passband with a gain of one and a stop band with a gain
of zero. In between, the filter drops off like a cosine function. When 𝛼 = 0, it
is a perfect LPF with a spectrum that is flat over the 1/T s bandwidth and zero
2.17 Raised-Cosine Filter
1.0
α = 0.0
α = 0.5
α = 1.0
H (f) 0.5
0.0
1
−
Ts
−
1
2Ts
Frequency (Hz)
1
2Ts
1
Ts
Figure 2.30 Spectrum of a raised-cosine filter.
elsewhere. Figure 2.30 shows the frequency response of a raised-cosine filter.
H RC (f ) = 0.5 for all values of 𝛼 at f = 1/2T s .
The impulse response is the inverse Fourier transform of (2.68) [22]:
( )
( ) cos 𝜋𝛼t
Ts
t
hRC (t) = sinc
(2.69)
( )2
Ts
1 − 2𝛼t
T
s
Low values of 𝛼 produce higher side lobes that extend into the adjacent pulses
hence increasing the probability of ISI. The time domain representation of the
raised-cosine pulse appears in Figure 2.31. Raised-cosine pulses are zero when
t = nT s for n ≥ 1.
Figure 2.32 shows a series of raised-cosine pulses spaced T s = 1 μs apart. The
peak of one pulse occurs at the nulls of the others. Individual pulses are graphed
on the left while their sum is graphed on the right. When 𝛼 = 0.0, pulse sidelobes
Figure 2.31 Raised-cosine
filter in time domain.
α = 0.0
α = 0.5
α = 1.0
hRC(t)
−4Ts −3Ts −2Ts −Ts
Ts
2Ts 3Ts 4Ts
55
2 Signals and Bits
Individual pulses
Combined pulses
α = 0.0
1
Pulses (V)
Pulses (V)
1
0.5
0
α = 0.0
0.8
0.6
0.4
0.2
0.0
α = 0.5
1
Pulses (V)
Pulses (V)
1
0.5
0
α = 0.5
0.8
0.6
0.4
0.2
0.0
α = 1.0
1
Pulses (V)
1
Pulses (V)
56
0.5
0
−4 −3 −2 −1
α = 1.0
0.8
0.6
0.4
0.2
0
1
2
3
4
0.0
−4 −3 −2 −1
t (μs)
0
1
2
3
4
t (μs)
Figure 2.32 Series of raised-cos pulses spaced T s = 1 μs apart.
are high, so the sum of the pulses shows significant distortion. Increasing 𝛼
decreases the pulse sidelobes and the ISI in their sum.
Example
One way to implement a raised-cosine channel transfer function is to have a
root raised-cosine filter (RRC) that has a frequency response of
√
(2.70)
HRRC (f ) = HRC (f )
at the transmitter and receiver, so the product of the two is a raised-cosine filter.
Redo Figures 2.30 and 2.31 for the RRC filter.
Solution
The impulse response for the RRC filter is
]
[
]
[
( ) sin 𝜋(1−𝛼)t + 4𝛼 t cos 𝜋 t (1 + 𝛼)
Ts
Ts
Ts
t
hRRC (t) = sinc
[
( )2 ]
Ts
𝜋 Tt 1 − 4𝛼t
T
s
(2.71)
s
Figure 2.33 is a plot of the spectrum and Figure 2.34 is a plot of the impulse
response.
2.18 Equalization
1.0
α = 0.0
α = 0.5
α = 1.0
H (f ) 0.5
0.0
1
−
Ts
−
1
2Ts
Frequency (Hz)
1
2Ts
1
Ts
Figure 2.33 Spectrum of a root-raised-cosine filter.
Figure 2.34 Root-raisedcosine filter in time domain.
α = 0.0
α = 0.5
α = 1.0
hRRC(t)
−4Ts −3Ts −2Ts −Ts
Ts
2Ts 3Ts 4Ts
2.18 Equalization
When a high-speed data signal propagates through a channel that has less
bandwidth than the signal (lossy channel), it becomes dispersed and picks
up noise, so the BER increases. Sharp symbol edges become rounded and
spread in time which in turn causes ISI as shown in Figure 2.35a. The zero
crossing times depend on the bit amplitudes and lead to significant jitter. In
the frequency domain, the lossy channel attenuates high-frequency content
to get the reduced spectrum in Figure 2.35b. An equalizer is a high-pass filter
(HPF) with a response that is the inverse of the channel response, so it will
completely recover the transmitted signal as illustrated in Figure 2.35c [29].
The infinite impulse response (IIR) filter in Figure 2.36 serves as a model for
most equalizers. The output of the IIR filter sampled at a period T s approximates the transmitted signal given by
ŝt (t) =
M
∑
m=0
bm sr (t − mT s ) +
N
∑
n=1
an ŝt (t − nT s )
(2.72)
57
58
2 Signals and Bits
Figure 2.35 Dispersion and equalization in the time domain. (a) Dispersion in a lossy
channel, (b) the channel attenuates high-frequency components, and (c) a HPF serves as an
equalizer. Source: Reprinted by permission of [30]; © 2017 IEEE.
FIR filter
sr (t)
b0
Σ
Σ
sr (t)
Ts
Ts
b1
a1
Ts
Ts
b2
a2
Ts
Ts
bM
aN
IIR filter
Figure 2.36 Equalizing filter.
2.18 Equalization
where t is sampled. If all the an = 0 then the filter becomes an all zero filter called
a finite impulse response (FIR) filter. It only has feed forward paths. When the
feedback is not zero, then the transfer function has poles and the filter becomes
IIR. The FIR filter has linear phase and is stable, while the IIR filter has nonlinear
phase, and its stability depends on the weights an and bn .
A zero-forcing equalizer [33] has a transfer function that is the inverse of the
channel transfer function.
1
Heq (f ) =
(2.73)
Hc (f )
This FIR filter cancels the ISI, but also amplifies the channel noise at the higher
frequencies, because all high-frequency components (signal and noise) get
amplified. An FIR filter becomes a good approximation of the inverse of the
channel transfer function when the weights, bm , (an = 0) are chosen to force
the system impulse response to zero at all the delayed samples (m > 0).
⎡ sr [0]
⎢ s [T ]
⎢ r s
⎢
⎢
⎣sr [MT s ]
sr [−Ts ]
···
sr [0]
⋱
···
sr [−MT s ]⎤ ⎡ b0 ⎤ ⎡1⎤
⎥ ⎢ b ⎥ ⎢0⎥
⋮
⎥⎢ 1⎥ = ⎢ ⎥
⎥ ⎢ ⋮ ⎥ ⎢⋮⎥
⎥⎢ ⎥ ⎢ ⎥
sr [0] ⎦ ⎣bM ⎦ ⎣0⎦
(2.74)
which in matrix form is
(2.75)
Sr w = hc
and the weights are found by
wopt = S−1
r hc
(2.76)
Example
Figure 2.37 is an example of a received pulse that [enters an equalizer. Find ]the
weights
for a zero-forcing filter
[
] if the samples are 0 0 0.1 0.85 −0.3 0.2 0 at
−3Ts −2T −T 0 T 2T 3T .
sr(t)
Figure 2.37 Received pulse example.
1.0
−3Ts −2Ts
−Ts
Ts
2Ts
3Ts
t
59
60
2 Signals and Bits
Solution
[
]
The ideal filter output for a two tap filter would be 1 0 0 . The coefficients are
found using (2.74)
⎡ 0.85
⎢
⎢−0.3
⎢ 0.2
⎣
0.1
0.85
−0.3
0 ⎤ ⎡b0 ⎤ ⎡1⎤
⎥⎢ ⎥ ⎢ ⎥
0.1 ⎥ ⎢b1 ⎥ = ⎢0⎥
0.85⎥⎦ ⎢⎣b2 ⎥⎦ ⎢⎣0⎥⎦
Solving for the weights produces b0 = 1.128, b1 = 0.412, b2 = −0.120.
Transmitting a training sequence through the channel to the equalizer at the
receiver produces the elements in the Sr matrix in (2.75). Narrow pulses or
pseudo random noise (PRN) make good training sequences with a full spectral
content across the bandwidth. PRN looks like noise but is a deterministic signal
that repeats with a very long period. It has a larger average power resulting in a
higher SNR for the same peak transmitted power of a narrow pulse, so the PRN
is usually used. Preset equalization periodically updates the filter weights due
to changes in the channel.
An alternative approach, the minimum mean square error (MMSE) equalizer,
reduces the mean-square error (MSE) between the equalized and transmitted
signal which in turn reduces both ISI and noise. The error is the difference
between the estimated data symbol at the equalizer output and the desired data
symbol
𝜀(t) = ŝt (t) − st (t)
(2.77)
where ŝt (t) is the equalizer output and st is the original transmitted signal. The
equalizer minimizes the average square of this error through an appropriate set
of weights, bn .
The error in (2.77) is a random variable, since the received signal is a random
variable that changes with time. The expected value of the square of the error is
where
E{|𝜀(t)|2 } = E{|st (t) − wT sr (t)|2 }
(2.78)
[
]T
sr (t) = sr [0] sr [Ts ] · · · sr [MT s ]
(2.79)
Expanding the right side leads to
E{|𝜀|2 } = E{|st (t)|2 } − 2w† E{st (t)sr (t)} + E{w† sr (t)s†r (t)w}
(2.80)
where superscript “†” indicates the complex conjugate transpose. For a zero
mean signal, the first term equals the signal variance which is a constant. Dropping the expected value, (2.80) becomes
E{|𝜀|2 } ≈ 𝜎s2 − 2w† E{st (t)sr (t)} + w† Cw
(2.81)
2.18 Equalization
where C is the covariance matrix given by
†
E{s†r (0)sr (Ts )}
···
E{s†r (0)sr (MT s )} ⎤
⎡ E{sr (0)sr (0)}
⎢ E{s† (T )s (0)}
⎥
E{sr (Ts )s†r (Ts )}
⋮
r
s r
⎥
C=⎢
⎢
⎥
⋱
⎢ †
⎥
†
†
⎣E{sr (MT s )sr (0)} E{sr (MT s )sr (Ts )} · · · E{sr (MT s )sr (MT s )}⎦
(2.82)
Now, take the derivative of (2.81) with respect to the weights and set it to zero
to find the optimum solution.
E{∇w |𝜀|2 } = −2st (t)E{s∗r (t)} + 2Cw∗opt = 0
(2.83)
Solving this equation for the optimum weights results in the Wiener–Hopf
solution:
wopt = C−1 st (t)E{sr (t)}
(2.84)
Solving for the optimum weights requires knowing st (t) and sr (t). Training
sequence or predefined sequence of symbols lead to estimates of sr (t).
Transmitting a training sequence has several drawbacks:
1. The spectral efficiency goes down, because nonmessage bits increase, thus
reducing the data rate.
2. To maintain an acceptable spectral efficiency, the training sequence is short,
so the filter does not average enough to significantly reduce noise.
3. If the channel changes after the training sequence is received, then the filter
weights are outdated and the BER increases.
Blind equalization omits the training sequence but exploits the statistical properties of the transmitted signal to estimate the channel and data. The weights
adjust until the statistical properties of ŝt (t) match the statistical properties of
st (t).
In mobile communications, the channel fading changes over time, so equalizers must quickly adapt to these time varying characteristics. An adaptive equalizer continuously changes the weights in order to keep up with a fast changing
channel [32]. Many adaptive algorithms exist to update the weights so as to minimize the SNR. A sampling of these algorithms are presented in later chapters.
Linear equalization faces three problems [30]:
1. Lossy channels require significant amplification of high frequency components that in turn significantly amplifies high-frequency noise and corrupts
the data.
2. The high amplification requires multiple stages with each stage limiting the
bandwidth and requiring considerable power.
61
62
2 Signals and Bits
FIR
filter
sr (t)
Σ
Limiter
st (t)
Figure 2.38 DFE equalizer.
Feedback
filter
Flip flop
sr (t)
Σ
Ts
st (t)
Figure 2.39 Operation of a
one-delay DFE.
Limiter
a
3. Cannot compensate for sharp notches in the channel transfer function, like
impedance discontinuities (mismatches) resulting from connectors and
other physical interfaces between boards, cables, etc.
Nonlinear equalization with an IIR filter overcomes these problems with feed
forward (bn ) and feedback (an ).
A decision-feedback equalizer (DFE) is a nonlinear equalizer that uses previously detected symbols as feedback to eliminate ISI on current symbols [31]
(Figure 2.38). The FIR and feedback filters can take various forms, for example
zero-forcing or MMSE. A major drawback of a DFE occurs in low SNR channels where the BER is relatively high. In that case, errors in the feedback quickly
cause the overall output error to rise.
Figure 2.39 explains the operation of a simple one delay DFE [30]. Feedback
is amplified then subtracted from the received signal. This combination is then
delayed by one symbol period. A limiter clips the signal peak in order to eliminate amplitude noise. A flip flop often replaces the combined time delay and
limiter. The resulting output looks very similar to the transmitted signal. ISI
and noise are dramatically reduced.
Problems
2.1
Record a voice signal for 5 s. Load the signal into MATLAB and plot it
like the one in Figure 2.1.
2.2
Plot the PSD of a 5 s voice signal (sig) in MATLAB. Approximate
the Fourier transform with the FFT (fast Fourier transform) using
sf=abs(fft(sig)). Your plot should be similar to the one in Figure 2.2.
Problems
2.3
Plot the spectrogram of a 5 s voice signal in MATLAB using the command that generated Figure 2.4.
2.4
What is the ASCII representation of the message: (a) “three,” (b) “radar,”
(c) “summer”?
2.5
A data source has five symbols that occur with the following frequencies:
Symbol
Frequency
A
24
B
12
C
10
D
8
E
8
Build and label a Huffman code tree.
2.6
A data source has seven symbols that occur with the following frequencies:
Symbol
a
Frequency
37
b
18
c
29
d
13
e
30
f
17
g
6
Build and label a Huffman code tree.
2.7
Write MATLAB code to find the Hamming distance and Hamming
weight. Use the strings a = [1 : 7 0 2] and b = [1 : 9] to demonstrate
your code.
2.8
Is the code [01 100 101 1110 1111 0011 0001] a prefix code? Is it optimum?
63
64
2 Signals and Bits
2.9
2.10
Find the bandwidth of Ps = cos2 [2𝜋(f − 2)] where 1.6 ≤ f ≤ 2.4 GHz.
Find the energy and power of
t < −1
⎧0
⎪
s(t) = ⎨2
−1 ≤ t ≤ 0
⎪ −t∕2
t>0
⎩2e
{
0 t≤0
s(t) =
1 t>0
s(t) = sinc(t)
2.11
Find the SNR of a 20 ms rectangular pulse with Gaussian noise
(𝜎 noise = 0.000 01 V) sampled at 10 kHz for 2 s.
2.12
A signal has an amplitude of 1 V in the presence of Gaussian noise that
has an RMS amplitude of 0.5 V. Find the SNR in dB.
2.13
A signal with power 3 mW has noise with power 0.001 mW. What is the
SNR in dB?
2.14
Find the SNR when the RMS signal level is 0.707 V, temperature is 300 K,
and bandwidth is 50 MHz.
2.15
Show the steps needed to go from (2.28) to (2.29).
2.16
What should the input SNR be for an output SNR = 10 dB in a receiver
with NF = 6 dB and B = 0.1 MHz?
2.17
An antenna is connected to an LNA (NF = 7 dB, G = 20 dB) via a cable
(1.5 dB attenuation). The output from the LNA is down converted by a
mixer (NF = 8 dB, G = 8 dB). The output passes through another amplifier (NF = 6 dB, G = 60 dB).
(a) Find the NF of the entire system.
(b) Find the NF of the entire system if the RF amplifier is placed before
the cable with 1.5 dB loss.
(c) Which system is better why?
2.18
A cascade of amplifiers starts with A and ends with (a) C and (b) B.
Problems
Amplifier
Gain (dB)
NF (dB)
A
4
2.3
B
12
3
C
20
6
2.19
An RF front end with G = 20 dB and NF = 6.5 dB feeds an ADC with
G = 0 dB and NF = 24 dB. Find the system NF.
2.20
Find the SNR of an ADC with (a) 10 bits, (b) 14 bits, and (c) 16 bits.
2.21
Find the capacity of an AWGN channel transmit 10 W at 1 MHz, and
power spectral density of noise of 2 × 10−9 W/Hz.
2.22
An analog signal with a bandwidth of 4 kHz is sampled at 1.25 times the
Nyquist frequency. Each sample is quantized into 256 levels with equal
probability.
(a) What is the information rate?
(b) Is it possible to transmit error-free signals over a channel having
AWGN with a 10 kHz bandwidth and SNR = 20 dB.
(c) Find the SNR needed for error-free transmission with AWGN having
a 10 kHz bandwidth and SNR = 20 dB.
(d) Compute the required bandwidth to transmit error-free signals over
an AWGN channel for a SNR = 20 dB.
2.23
Plot C from (2.52) vs. SNR (−10 dB ≤ SNR ≤ 20 dB) and B (1 kHz ≤ B ≤
1 GHz).
2.24
How much time is needed to test a signal for a BER of 10−11 , 10−13 , 10−15
for data rates (bits/s) 39.813 12 Gbps, 2.488 32 Gbps, 155.52 Mbps?
2.25
Find both the LRC and VRC for the ASCII representation of the word
(a) spring, (b) radar, and (c) snowing.
2.26
Find the encoded data using CRC parity bits for the data [100100] for the
generating polynomial x3 + x + 1. Check the received data to see if it is
error free. Assume that the received word is 100000001. Is this reception
error free?
2.27
Find the CRC parity bits for the data [11100110] for the generating polynomial x4 + x3 + 1.
65
66
2 Signals and Bits
2.28
Find (a) the 16 possible code words and (b) H. Given
⎡1
⎢0
G=⎢
0
⎢
⎣0
0
1
0
0
0
0
1
0
0
0
0
1
0
1
1
1
1
0
1
1
1⎤
1⎥
0⎥
⎥
1⎦
2.29
Given the data bits [1011], find the parity bits for the convolutional code
given g 1 = [111] and g 2 = [110]. Assume that all previous bits are 0.
2.30
A message contains the bits [1011]. If
⎡1 1 0 1 1
H = ⎢1 0 1 1 0
⎢
⎣0 1 1 1 0
0
1
0
0⎤
0⎥
⎥
1⎦
then find (a) the code word, (b) the syndrome if the received bits are
[1010010], and (c) what bit is in error due to the syndrome in (b)?
2.31
Use MATLAB to plot (2.68) for 𝛼 = 0, 0.5, and 1.0.
2.32
Use MATLAB to plot (2.69) for 𝛼 = 0, 0.5, and 1.0.
2.33
Make a plot of R vs. 𝛼 for select values of B between 1 MHz and 10 GHz.
2.34
A baseband signal has four levels with T s = 100 μs.
(a) If the root raised-cosine filter has 𝛼 = 0.3, what is the minimum bandwidth?
(b) How long does it take to transmit 1 000 000 bits?
(c) How many symbol states are needed to transmit the information in
half the time given the same bandwidth?
2.35
A satellite receiver operating at 2 GHz has an LNA at 127 K and 20 dB
gain followed by an amplifier with F = 12 dB and a gain of 80 dB. Find
the system noise factor and T eq .
2.36
The first amplifier has a gain of 12 dB and a noise temperature of 28 K.
The second amplifier has a noise temperature 200 K. What is T eq ?
2.37
An amplifier has a noise temperature of 60 K. What is its noise figure and
noise factor?
2.38
An amplifier has noise figure of 1.5 dB. What is its noise temperature?
References
2.39
Determine the increase in the total noise figure (NF) if an input amplifier
with a noise figure of 3 dB is replaced with an amplifier with a noise figure
of 6 dB.
2.40
An overall noise figure (NF) of 13 dB indicates that the noise level
increases by a factor of what as compared to the increase in signal level?
2.41
Find the weights for a zero-forcing filter if the samples are [0 0.2 0.45
1.0 0.3 −0.25 0] at [−3Ts −2T −T 0 T 2T 3T].
2.42
Find the weights for a zero-forcing filter if the samples are [−0.08 0.25]
1 0.3 −0.06] at [−2T −T 0 T 2T].
References
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14 Peterson, W.W. and Brown, D.T. (1961). Cyclic codes for error detection.
Proceedings of the IRE 49 (1): 228–235.
15 Elias, P. (1955). Coding for noisy channels. IRE Convention Record 4: 37–46.
16 Peterson, W.W. and Weldon, E.J. (1971). Error Correcting Codes. In: MIT
Press, Revised 2e.
17 Viterbi, A.J. (1967). Error bounds for convolutional codes and an asymp-
18
19
20
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22
23
24
25
26
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29
30
totically optimum decoding algorithm. IEEE Transactions on Information
Theory 13 (2): 260–269.
https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/
6-02-introduction-to-eecs-ii-digital-communication-systems-fall-2012/
lecture-slides/MIT6_02F12_lec06.pdf (accessed 14 August 2017).
Berrou, C., Glavieux, A., and Thitimajshima, P. (1993). Near Shannon limit
error-correcting coding and decoding: turbo-codes. 1. In: IEEE International Conference on Communications, 1993. ICC ’93 Geneva. Technical
Program, Conference Record, Geneva, vol. 2, 1064–1070.
Burr, A. (2001). Turbo-codes: the ultimate error control codes? Electronics
& Communication Engineering Journal 13 (4): 155–165.
Shi, Y.Q., Xi, M.Z., Ni, Z.-C., and Ansari, N. (2004). Interleaving for combating bursts of errors. IEEE Circuits and Systems Magazine 4 (1): 29–42,
First Quarter.
Couch, L.W. II, (1993). Digital and Analog Communication Systems, 4e.
New York: Macmillan Publishing Co.
Reed, I.S. and Solomon, G. (1960). Polynomial codes over certain finite
fields. Journal of the Society for Industrial and Applied Mathematics 8 (2):
300–304.
Ludwig, R. and Taylor, J.(2002). Voyager telecommunications, JPL DESCANSO Design and Performance Summary Series, Pasadena, CA.
http://wireless.ece.ufl.edu/twong/Notes/Comm/ch4.pdf (accessed 26 June
2018).
D. O. North, "An Analysis of the factors which determine signal/noise discrimination in pulsed-carrier systems," in Proceedings of the IEEE, vol. 51,
no. 7, pp. 1016-1027, July 1963; originally published as RCA Laboratories
Report PTR-6C, Jun 1943.
Turin, G. (1960). An introduction to matched filters. IRE Transactions on
Information Theory 6 (3): 311–329.
Scholtz, R. (1993). Multiple access with time-hopping impulse modulation.
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Record. Communications on the Move., IEEE, Boston, MA, vol. 2, 447–450.
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wireless communication and its applications. International Journal of Industrial Electronics and Electrical Engineering: 83–86.
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IEEE Solid-State Circuits Magazine 9 (4): 13–132, Fall.
References
31 Austin, M.E. (1967). Decision-Feedback Equalization for Digital Communi-
cation Over Dispersive Channels. Tech. Rep. 437, Lincoln Laboratory.
32 Lucky, R.W. (1966). Techniques for adaptive equalization of digital commu-
nication systems. The Bell System Technical Journal 45 (2): 255–286.
33 Lucky, R.W. (1965). Automatic equalization for digital communication. The
Bell System Technical Journal 44 (4): 547–588.
69
71
3
Passband Signals
Information starts as an analog baseband signal with its lowest frequency close
to zero, such as audio, video, sensors, images, etc. Passband signals, on the other
hand, are baseband signals elevated to a higher frequency in order to fit into
particular slots in the spectrum. Figure 3.1 shows a baseband signal modulated
to a passband signal then frequency multiplexed with two other signals, so that
they all share the spectrum without interfering with each other. Consider the
case of amplitude modulation (AM) radio. A low-pass filtered audio signal has
a spectrum from 0 Hz to 10 kHz. Then, a modulator upconverts this signal to
a higher frequency that fits into the 20 kHz channel assigned to the AM station. Frequency multiplexing fits all the stations into the spectrum assigned to
AM broadcasting. The receiver down-converts or demodulates the signal back
to baseband for listening. This chapter introduces different analog and digital
modulation schemes as well as several approaches to multiplexing.
3.1 Carrier
The carrier is a single frequency electromagnetic wave (harmonic or tone) represented in rectangular coordinates as:
⃗ = x̂ Ex cos(2𝜋fc t − kx x) + ŷ Ey cos(2𝜋fc t − ky y + 𝜓y )
E(t)
+ ẑ Ez cos(2𝜋fc t − kz z + 𝜓z )
(3.1)
where
f c = carrier frequency
t = time
c = 𝜆/T = speed of light
𝜆 = wavelength
T = time period
√
k=
kx2
+
ky2
+
kz2
=
√(
2𝜋
𝜆x
(
)2
+
2𝜋
𝜆y
(
)2
+
2𝜋
𝜆z
)2
=
2𝜋
= wavenumber
𝜆
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
72
3 Passband Signals
Baseband
Passband
Modulation
f
f
f
Modulation
and
multiplexing
f
f
f
Figure 3.1 Modulation converts the baseband signal to a passband signal. Multiplexing
inserts it into the proper place in the spectrum to avoid interference with other signals.
𝜆x , 𝜆y , 𝜆z = wavelength projected in the x, y, and z directions
x̂ , ŷ , and ẑ = unit vectors in the x, y, and z directions
Ex , Ey, and Ez = magnitudes of the electric fields in the x, y, and z directions
𝜓 y and 𝜓 z = phases of the y and z components relative to the x component
A carrier contains no information but serves to transport the baseband information signal from the transmitter to the receiver at the desired passband frequencies.
The electric field in (3.1) can be written as:
⃗ = Re{Eej2𝜋fc t }
E(t)
(3.2)
where E is the complex steady-state phasor (time independent):
̂ x e−jkx x + ŷ Ey e−jky y ej𝜓y + ẑ Ez e−jkz z ej𝜓z
E = xE
(3.3)
Since this chapter only deals with the time-varying part of (3.2), E is ignored.
3.2 Amplitude-Modulated Signals
AM changes the carrier amplitude in sync with the amplitude of the message
signal, m(t). A sinusoidal carrier, for example V c cos(2𝜋f c t), multiplies the
message to produce double sideband (DSB) modulation:
s(t) = m(t)Vc cos(2𝜋fc t)
(3.4)
Assume the message signal is a single harmonic at f m with amplitude V m /V c
m(t) =
Vm
cos 2𝜋fm t
Vc
(3.5)
3.2 Amplitude-Modulated Signals
Substituting (3.5) into (3.4) yields
s(t) = Vm cos(2𝜋fm t) cos(2𝜋fc t)
V
V
= m cos(2𝜋( fc − fm )t) + m cos(2𝜋( fc + fm )t)
2
2
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
lower sideband
(3.6)
upper sideband
As seen in (3.6), the resulting signal has two frequency bands on either side
of f c but the carrier disappeared, hence the name double sideband modulation.
The lower sideband lies f m below the carrier, and the upper sideband lies f m
above the carrier. Figure 3.2 shows how ±m(t) bounds the modulated carrier
signal when V m = V c = 1.0 V and f c = 300 and f m = 20 kHz.
DSB modulation requires synchronized detection (transmitter and receiver
aligned in frequency and phase). A local oscillator (LO) in the receiver reproduces an exact copy of the transmit carrier that multiplies the received signal.
This multiplication results in a replica at baseband and another replica at twice
the carrier frequency. A low pass filter (LPF) discards the replica at 2f c but
keeps the desired baseband signal. Figure 3.3 diagrams the process as well as
shows the mathematical derivation.
Figure 3.2 DSB AM signal.
3
2
s(t)
1
0
−1
2
−3
0
20
40
60
t (μs)
m(t) cos(2πfct + φ) =
1
1
m(t) cos φ + m(t) cos(4πfct + φ)
2
2
m(t)
LPF
cos(2πfct + φ)
Figure 3.3 Synchronous detection of a DSB signal.
1
m(t)cos φ
2
80
100
120
73
74
3 Passband Signals
Adding a carrier to (3.6) produces an AM signal that does not require synchronous detection.
(3.7)
s(t) = Vc [1 + m(t)] cos(2𝜋fc t)
This modulated signal has an envelope with the same shape as the message.
As a result, the receiver consists of a simple envelope detector that recovers
m(t), unlike DSB modulation. Small variations in the carrier frequency do not
impact signal reception. An AM modulation of the message signal in (3.5) has
the following representation:
s(t) = Vc [1 + m(t)] cos(2𝜋fc t)
= Vc cos(2𝜋fc t) + Vm cos(2𝜋fm t) cos(2𝜋fc t)
V
V
= Vc cos(2𝜋fc t) + m cos(2𝜋(fc − fm )t) + m cos(2𝜋(fc + fm )t) (3.8)
2
2
⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
carrier
lower sideband
upper sideband
Figure 3.4 shows the AM spectrum with a carrier and two copies of the message at the harmonics at f c ± f m . Adding the carrier takes power away from
the message signal which in turn decreases the signal to noise ratio (SNR). The
components of (3.8) appear in the time-frequency plot in Figure 3.5.
An AM envelope detector has a diode that passes only the positive portion
of the waveform then a shunt capacitor that strips off the high-frequency components and leaves the baseband signal. Figure 3.6 shows an AM signal passing
through an envelope detector. This asynchronous detection does not require
an LO to recover the baseband signal.
Applying Euler’s identity to (3.8) produces a complex AM spectrum.
]
V
V [
s(t) = c [ej2𝜋fc t + e−j2𝜋fc t ] + m ej2𝜋 (fc −fm )t + e−j2𝜋 (fc −fm )t
2
4
]
Vm [ j2𝜋 (f +f )t
e c m + e−j2𝜋 (fc +fm )t
+
(3.9)
4
Figure 3.4 AM spectrum.
Amplitude
Vc
Vm/ 2
fc−fm
fc
fc + fm
f
3.2 Amplitude-Modulated Signals
Amplitude
fm
fc−fm
fc+fm
fc
f
Up
Ca
pe
Lo
Me
rrie
rs
we
ss
r
ide
r
ag
s
ide
ba
e
t
ula
nd
ba
ted
nd
sig
na
l
Mo
d
Figure 3.5 Time–frequency plot of an AM signal components.
Figure 3.6 Envelope
demodulation of an AM signal.
This equation may be rewritten as
]
[
)
Vc j2𝜋f t
𝛽AM ( j2𝜋f t
−j2𝜋fm t
c
m
s(t) =
+e
1+
e
e
2
2
[
]
)
Vc −j2𝜋f t
𝛽AM ( j2𝜋f t
−j2𝜋fm t
c
m
+ e
+e
1+
e
2
2
(3.10)
where the AM modulation index is
V
𝛽AM = m
Vc
(3.11)
Figure 3.7 shows the complex spectrum for the AM signal in (3.9). The spectral components of the one-sided spectrum in Figure 3.4 split into positive- and
negative-frequency components with half the amplitude.
Amplitude
Vc / 2
Vm / 4
−fc−fm −fc −fc + fm
Figure 3.7 AM complex spectrum.
fc − fm fc
fc + fm
f
75
3
3
2
2
1
1
s(t)
s(t)
3 Passband Signals
0
0
−1
−1
−2
−2
−3
0
20
40
60
t (μs)
80
100
−3
120
0
20
40
(a)
3
2
2
1
1
0
−1
−2
−2
0
20
40
60
t (μs)
80
100
120
80
100
120
0
−1
−3
60
t (μs)
(b)
3
s(t)
s(t)
76
80
100
120
−3
0
20
(c)
40
60
t (μs)
(d)
Figure 3.8 Effect of the modulation index on the time domain AM signal. (a) 𝛽 AM = 0.0. (b)
𝛽 AM = 0.5. (c) 𝛽 AM = 1.0. (d) 𝛽 AM = 2.0.
The modulation index determines the bounds of the AM signal envelope.
Figure 3.8 shows a plot of (3.10) for four different values of 𝛽 AM . When 𝛽 AM = 0,
only the carrier wave exists. When 0 < 𝛽 AM < 1.0, the carrier signal has higher
amplitude than the message signal, and the message signal clearly forms an
envelope that defines the carrier amplitude. Figure 3.8b shows the modulated
signal when 𝛽 AM = 0.5. When 𝛽 AM = 1.0, the positive envelope touches the
negative envelope (Figure 3.8c). Beyond this point, envelope detection does
not perfectly retrieve m(t). When 𝛽 AM > 1.0 (e.g. 𝛽 AM = 2.0 in Figure 3.8d),
over modulation occurs. Putting more power into the message than the carrier
makes sense, but the envelope of the modulated signal becomes distorted and
the carrier experiences a 180∘ phase shift. Overmodulation introduces spurious
harmonics into the signal that cause radiation outside the signal bandwidth.
The Fourier transform of (3.8) yields the frequency domain representation of
the complex AM spectrum.
S(f ) =
]
Vc [
𝛿(f − fc ) + 𝛿(f + fc )
2
]
V [
+ m 𝛿(f − fc + fm ) + 𝛿(f + fc − fm )
4
]
Vm [
+
𝛿(f − fc − fm ) + 𝛿(f + fc + fm )
4
(3.12)
3.2 Amplitude-Modulated Signals
The AM signal power at a resistor Re is
[ 2
]
Vm2
1 Vc
P=
+
Re 2
4
(3.13)
Substituting V m = 𝛽 AM V c into (3.13) yields
P=
1
2
[V 2 + 0.5𝛽AM
Vc2 ]
2Re c
(3.14)
Substituting the carrier power, Pc =
[
]
2
P = Pc 1 + 0.5𝛽AM
Vc2
2Re
, into (3.14) produces
(3.15)
Example
Use MATLAB to modulate a 300 kHz carrier with a voice signal using 𝛽 AM =
0.5.
Solution
Figure 3.9 shows a small part of a voice signal modulated with the 300 kHz carrier. The following code captured the voice signal into the vector "song" and
normalizes the amplitude. The key commands to find the modulated signal are
load music; % this loads the voice file "music" into
myRecording
song=myRecording'; % the vector "song" contains the music
song=song/max(song); % normalize the amplitude
t=[0:1250]*1e-6; % time in microseconds
car= cos(2*pi*300e3*t); % carrier
bAM=0.5; % modulation index
vAM=[1+bAM*song]*car; % AM signal
The AM radio frequency band extends from 535 to 1605 kHz as shown
in Figure 3.10. Carrier frequencies occur at 10 kHz intervals from 540 to
Figure 3.9 AM voice signal.
3
2
νAM
1
0
−1
−2
−3
0
200
400
600
t (μs)
800
1000
1200
77
3 Passband Signals
1590 kHz 1600 kHz
535 kHz
1605 kHz
540 kHz 550 kHz 560 kHz
Lower
sideband
10 kHz
Upper
sideband
Figure 3.10 The AM radio spectrum.
1185 kHz
Received power (dBm)
78
1195 kHz
−94
−98
−102
−106
535
1070
Frequency (kHz)
1605
Figure 3.11 Measured broadcast AM spectrum near Boulder, CO with expanded view from
1185 to 1195 kHz.
1600 kHz. Two sidebands and the carrier for transmitting voice signals lie
within the 10 kHz bandwidth. The measured broadcast AM spectrum from
1185 to 1195 kHz near Boulder, CO appears in Figure 3.11. The inset box
shows an expanded frequency view about the 1190 kHz carrier frequency
(1185–1195 kHz) where the radio station resides. Note the sidebands on either
side of the carrier.
AM is not efficient, because the carrier consumes power, and the information resides in both the upper and the lower sidebands. Single sideband (SSB)
modulation increases efficiency by only transmitting one sideband of AM
(usually the upper sideband). This approach halves the AM bandwidth while
still transmitting the entire message signal. SSB suppressed carrier modulation
removes the carrier to reduce the transmitted power. In order to make the
signal easier to detect, many SSB schemes have some remnant of the carrier
for receiver synchronization. Simple envelope detection does not work for SSB
signals.
Digital AM or amplitude shift keying (ASK) represents binary symbols by
discrete amplitude steps. The simplest form of ASK have two states: the presence of a carrier wave that indicates a “1,” and its absence that indicates a “0.”
3.2 Amplitude-Modulated Signals
1
m(t)
(a)
0.5
0
1
0
0
1
0
1
1
(b)
Vc cos(2π fct)
0
1
0
−1
1
s(t)
(c)
0
−1
1
m(t)
(d)
0.5
01
0
00
10
11
1
s(t)
(e)
0
−1
0
10
20
30
40
50
60
70
t(ns)
Figure 3.12 ASK modulation. (a) 8 bits, (b) carrier, (c) OOK, (d) symbols with 2 bits, and (e)
4-ASK.
Figure 3.12a shows an on–off keying (OOK) bit stream multiplying a carrier in
Figure 3.12b to get the ASK signal in Figure 3.12c. Using more than two amplitude states to represent symbols requires a high SNR in order to distinguish the
different symbols. M-ASK modulation uses M = 2Nbits amplitude levels to represent symbols that have N bits bits. Figure 3.12d shows four different symbols
that multiply the carrier to get the 4-ASK signal in Figure 3.12e. The measured
power received as a function of frequency for a 2 GHz ASK signal with a symbol
rate of 200 kbps appears in Figure 3.13. The carrier causes the spike at 2 GHz.
AM has the advantages of being simple and cheap to implement. The disadvantages include low power efficiency, low bandwidth efficiency, and high
sensitivity to noise. As a result, AM works well for narrowband applications
that tolerate noise as in the transmission of audio signals. AM consumes a lot of
spectrum and over half of its power lies with the carrier which has no information. DSB does not waste power in a carrier but has the same bandwidth as AM
and requires synchronous detection. SSB modulation has half the bandwidth of
AM, puts all power in one information band, has less noise due to smaller bandwidth, and prevents selective fading (carrier and sidebands arriving at different
times) but requires synchronous detection.
79
3 Passband Signals
Received power (dBm)
80
−50
−70
−90
−110
1.9975
2
Frequency (GHz)
2.0025
Figure 3.13 Measured spectrum of a 2-ASK signal with a carrier frequency of 2 GHz and a
symbol rate of 200 kbps.
3.3 Frequency-Modulated Signals
Frequency modulation (FM) varies the frequency of the carrier in proportion
to m(t). The voltage waveform of an FM signal looks like
[
]
t
s(t) = Vc cos 2𝜋fc t + 2𝜋Δf
m(𝜏)d𝜏
(3.16)
∫0
where Δf is the maximum frequency deviation from f c . As in AM, assume m(t)
is a single harmonic at f m
m(t) = Vm cos 2𝜋fm t
(3.17)
Substituting m(t) into (3.16) and integrating produces
s(t) = Vc cos[2𝜋fc t + 𝛽FM sin(2𝜋fm t)]
(3.18)
where the FM modulation index is
𝛽FM =
Δf
fm
(3.19)
and Δf ∝ Am .
Example
Given a single harmonic message signal at 1 GHz and a carrier at 10 GHz, plot
the FM modulated signal for half a period of the message when 𝛽 FM = 0.2, 1.0,
2.0, and 5.0 and V c = V m = 1V .
Solution
Figure 3.14 shows the plots of s(t) = cos[2𝜋107 t + 𝛽 FM sin(2𝜋106 t)].
3.3 Frequency-Modulated Signals
1
s(t)
(a)
Message
0
Carrier
−1
1
s(t)
(b)
0
−1
1
s(t)
(c)
0
−1
1
s(t)
(d)
0
−1
1
s(t)
(e)
0
−1
0
0.2
0.4
0.6
0.8
1.0
t (ns)
Figure 3.14 FM modulation with f c = 10 GHz and f m = 1 GHz. (a) Message signal
superimposed on carrier, (b) 𝛽 FM = 0.2, (c) 𝛽 FM = 1.0, (d)𝛽 FM = 2.0, and (e) 𝛽 FM = 5.0.
Rewriting (3.18) in terms of Bessel functions results in an expression having
an infinite number of harmonics in the FM spectrum.
s(t) = Vc
∞
∑
[
]
Jn (𝛽FM ) cos 2𝜋(f + nf m )c t
(3.20)
n=−∞
where J n (𝛽 FM ) is an nth order Bessel function evaluated at 𝛽 FM (Figure 3.15).
Next, taking the Fourier transform of (3.20) results in
∞
)[
]
Vc ∑ (
𝛿(f − fc − nf m ) + 𝛿(f + fc + nf m )
J 𝛽
S(f ) =
2 n=−∞ n FM
Figure 3.15 Bessel functions
of the first kind.
Jn(βFM)
1
J0
J1
0.5
J2
(3.21)
J3
0
−0.5
0
2
4
βFM
6
8
81
82
3 Passband Signals
βFM = 0.2
fc
fc − fm fc + fm
βFM = 1.0
B
βFM = 2.0
B
βFM = 5.0
B
B
Figure 3.16 FM bandwidth as a function
of 𝛽 FM .
f
f
f
f
The peak amplitude of J n (𝛽 FM ) decreases with increasing n. As 𝛽 FM increases,
the number of effective sidebands in the FM signal increases, which in turn,
increases the bandwidth. When 𝛽 FM = 0.25 only one significant sideband exists.
Narrowband FM has a carrier, an upper sideband, and a lower sideband similar
to AM. Wideband FM occurs when 𝛽 FM > 1.0. If 𝛽 FM = 5, then the spectrum
has eight significant sidebands. Figure 3.16 shows the FM spectrum that correspond to the modulated signals in Figure 3.14.
An FM signal that varies in frequency from f c − Δf to f c + Δf has a bandwidth
given by Carson’s rule [1]:
)
)
(
(
(3.22)
B = 2 Δf + fm = 2 𝛽FM + 1 fm
where f m is the highest frequency in the modulating signal. In North America,
Δf = 75 kHz for commercial FM broadcasting in order to prevent adjacent stations from interfering with one another. FM radio station carrier frequencies
always end in 0.1, 0.3, 0.5, 0.7, or 0.9 MHz (88.1 MHz but not 88.0 MHz). If the
highest frequency in an audio signal is f m = 15 kHz, then 𝛽 FM = 5. The bandwidth is B = 2(5 + 1)5 kHz = 180 kHz which is close to the 200 kHz allocated
channel bandwidth. Figure 3.17 shows the measured broadcast FM spectrum
from 87.5 to 108.0 MHz near Boulder, CO. The inset box has an expanded frequency view at the 87.5 MHz carrier frequency of a radio spectrum.
Example
Use MATLAB to frequency modulate a 20 kHz carrier with a voice signal and
Δf = 10 kHz.
Solution
]
[
t
Substitute into (3.16) to get v(t) = cos 40𝜋t + 20𝜋 ∫0 m(𝜏)d𝜏 with t in ms.
Figure 3.18 is a MATLAB plot of the voice signal superimposed on the modulated signal.
3.3 Frequency-Modulated Signals
Received power (dBm)
88.35 MHz
88.65 MHz
−60.0
−70.0
−80.0
−90.0
87.5
97.75
Frequency (MHz)
108.0
Figure 3.17 Measured broadcast FM spectrum near Boulder, CO.
1
s (t)
0.5
0
−0.5
−1
0
0.2
0.4
0.6
0.8
1
1.2
t (ms)
Figure 3.18 Solid line is voice signal and dotted line is the FM-modulated carrier.
Digital FM, called frequency shift keying (FSK), represents symbols by distinct frequencies. Binary frequency shift keying (BFSK) has two possible waveforms:
{
(
)
Ac cos 2𝜋fhi t
for a"1"
(3.23)
s(t) =
(
)
for a"0"
Ac cos 2𝜋flo t
where f hi = f c + Δf and f lo = f c − Δf . Figure 3.19 shows a BFSK signal corresponding to a bit stream of 01001011 with each bit 1 μs long. In this example,
f lo = 3.9 MHz and f hi = 5.2 MHz. M-FSK (multiple frequency shift keying) uses
M different frequencies to represent M symbols, where each symbol has log2 M
bits. Figure 3.20 shows the power received as a function of frequency for a
2 GHz FSK signal with a symbol rate of 200 kbps.
FM has the advantage that amplitude variations due to noise do not degrade
signal detection. Most noise impacts the signal amplitude and not the frequency or phase. FM performance benefits from using nonlinear amplifiers
83
3 Passband Signals
0
1
0
0
1
0
1
1
s(t)
1
0
−1
0
1
2
3
4
5
6
7
8
t (µs)
Figure 3.19 BFSK signal.
Received power (dBm)
84
−60
−80
−100
−120
1.9975
2
2.0025
Frequency (GHz)
Figure 3.20 Measured spectrum of an FSK signal with a carrier frequency of 2 GHz and a
symbol rate of 200 kbps.
that have a higher efficiency than the AM linear amplifiers. FM has the
disadvantage of complicated modulation and demodulation circuits compared
to AM.
3.4 Phase-Modulated Signals
Phase modulation (PM) changes the carrier phase in proportion to m(t). A PM
signal takes the form
[
]
s(t) = Vc cos 2𝜋fc t + 𝛽PM m(t)
(3.24)
where 𝛽 PM (rad/V), the PM index, equals the peak phase deviation and
𝛽 PM ∝ V m . Note that (3.16) and (3.24) look very similar. In fact, FM falls into
the PM category. Carson’s rule also gives the bandwidth for PM signals as
B = 2(𝛽PM + 1)fm
(3.25)
3.4 Phase-Modulated Signals
0
s(t)
1
1
0
0
1
0
1
1
0
−1
0
10
20
30
50
40
60
70
80
t (ns)
Figure 3.21 BPSK where the 2 bit values are 180∘ out of phase.
In digital modulation, PM represents binary symbols with unique phase values. Binary phase shift keying (BPSK) has two phase states with a “1” 180∘ out of
phase with a “0.” For instance, a BPSK signal has the following representation:
(
)
⎧
for a"1"
⎪Ac cos 2𝜋fc t
s(t) = ⎨
(
)
⎪−Ac cos 2𝜋fc t
for a"0"
⎩
(3.26)
An example of an 8-bit BPSK signal appears in Figure 3.21. No phase change
occurs when consecutive bits remain the same. A constellation diagram displays symbol samples in the real-imaginary or in-phase-quadrature (IQ) plane.
Figure 3.22 shows the measured constellation and eye diagram plots for BPSK
as the transmitted signal decreases but the noise stays the same. The eye shrinks
as the SNR decreases relative to the noise. Ellipses encircle symbols in the constellation diagram. As the SNR decreases, the ellipses increase in size to the
point where they overlap. Distinguishing symbols becomes more and more difficult with decreasing SNR.
Quadrature phase shift keying (QPSK) maps four symbols containing 2 bits
each into four unique carrier phases. The phase state to symbol mapping is
given by
(
)
⎧Ac cos 2𝜋fc t − 450
⎪
(
)
⎪
0
⎪Ac cos 2𝜋fc t − 135
s(t) = ⎨
(
)
⎪Ac cos 2𝜋fc t − 3150
⎪
⎪A cos (2𝜋f t − 2250 )
⎩ c
c
for "00"
for "01"
(3.27)
for "10"
for "11"
85
86
3 Passband Signals
(a)
(b)
(c)
(d)
Figure 3.22 Measured constellation plot (left) and eye diagram (right) for BPSK as the
transmitted signal level decreases: (a) 10 dBm, (b) −35 dBm, (c) −40 dBm, and (d) −43 dBm.
Using the trigonometric identity cos(x + y) = cos x cos y − sin x sin y, (3.27)
becomes
)
(
)]
[
(
⎧
for "00"
⎪0.707Ac cos 2𝜋fc t + sin 2𝜋fc t
)
(
)]
[
(
⎪
for "01"
⎪0.707Ac − cos 2𝜋fc t + sin 2𝜋fc t
(3.28)
s(t) = ⎨
)
(
)]
[
(
for "10"
⎪0.707Ac cos 2𝜋fc t − sin 2𝜋fc t
)
(
)]
[
(
⎪
for "11"
⎪0.707Ac − cos 2𝜋fc t − sin 2𝜋fc t
⎩
If “cos” is “in-phase,” then “sin” is “quadrature” or 90∘ out-of-phase. Figure 3.23
shows a plot of the constellation diagram for (3.28). Arrows point from the
3.4 Phase-Modulated Signals
Q
Figure 3.23 IQ plot of a QPSK signal.
00
01
45°
I
10
11
0 1
s(t)
1
0 0
1 0
1 1
0
−1
0
10
20
30
40
t (ns)
50
60
70
80
Figure 3.24 QPSK with four phase changes for the four possible symbols.
present phase state to the next possible phase state. QPSK allows any of the four
phase states to follow the current phase state. The QPSK signal in Figure 3.24
has four symbols “01,” “00,” “10,” and “11.” Arrows start at the current state
and point at possible future states. The phase changes do not produce a smooth
transition between symbols in the time domain.
A QPSK signal has up to a 180∘ phase change between symbols (e.g. 00 → 11).
In a practical system, large changes in phase produce significant unwanted
spectral components outside the desired bandwidth. Alternative forms of
QPSK limit phase changes to less than 180∘ phase changes between states. As
an example, offset quadrature phase shift keying (OQPSK) delays the quadrature portion of the signal or the odd bits by 1 bit-period (half a symbol-period)
relative to the even bits, so that the in-phase and quadrature components
never change at the same time. This approach limits the maximum phase shift
between symbols to only 90∘ . Figure 3.25 shows an IQ plot of an OQPSK signal.
In this case, three instead of four phase states possibly follow the current phase
state. No symbol sequence has a path through the origin, which corresponds to
a 180∘ phase change. Generating the OQPSK symbols resemble QPSK, except
for the 1-bit time delay (T b ) in the quadrature channel shown in Figure 3.26.
87
88
3 Passband Signals
Q
01
Figure 3.25 IQ plot of an OQPSK signal.
00
I
11
10
×
cos(2πfct)
I(t)
Rb = 1 / Tb
Two bit
serial to
parallel
converter
∼
Σ
Rb /2
90°
Q(t)
−sin(2πfct)
OQPSK
delay Tb
×
Figure 3.26 QPSK and OQPSK modulation. OQPSK has a 1 bit time delay in the quadrature
path.
The I and Q channels have half the bit rate (Rb /2) of the input data (Rb ).
Figure 3.27 shows the output I and Q bits for QPSK vs. OQPSK when the input
symbols are “10,” “11,” “00,” “01,” and “11.” QPSK and OQPSK have identical
in-phase bits. In QPSK, the output phase changes as much as 180∘ with every
symbol. The 1-bit delay in the quadrature channel of OQPSK, on the other
hand, causes the output phase to change with every bit (twice as fast as QPSK).
𝜋/4-QPSK also reduces the phase change between symbols by using two
identical constellations with one rotated 45∘ relative to the other (Figure 3.28).
The four open circles in the constellation plot correspond to odd-numbered
symbols 1, 3, 5, …, while the closed circles correspond to even-numbered
symbols 2, 4, 6, . . . . In Figure 3.28, the message starts with symbol, 01, on
the open circle constellation. The second symbol, 00, is on the closed circle
constellation. The sequence moves to symbol, 10, then finally to 11. 𝜋/4-QPSK
has a maximum phase shift of 135∘ . A plot of the resulting 𝜋/4-QPSK signal
3.4 Phase-Modulated Signals
I
Input
Output phase
QPSK
Q
I
0
1
1
π
−
4
Q
I
Q
I
1
0
0
0
π
4
3π
−
4
Q
I
1
1
Q
1
π
4
3π
4
Q(t)
QPSK
I(t)
OQPSK
Q(t−Tb)
Output phase
OQPSK
−
π π
−
4 4
π
4
3π 3π 3π 3π
−
−
4
4
4 4
π
4
π
4
Figure 3.27 Plots of the in-phase (V I and, quadrature (V Q ) parts of the QPSK and OQPSK
signals.
Figure 3.28 IQ diagram for 𝜋/4 QPSK with the bit
sequence of 01 to 00 to 10 to 11.
Q
00
01
00
01
10
11
I
10
11
for the symbol sequence “01,” “00,” “10,” and “11” appears in Figure 3.29.
Demodulation of 𝜋/4-QPSK consists of differential detection in which the
phase of the previous symbol serves as a reference for the phase of the current
symbol. In the presence of multipath and fading, 𝜋/4-QPSK out performs
OQPSK [2].
Minimum shift keying (MSK) is the same as continuous-phase FSK with
𝛽 FM = 0.5. MSK has no phase discontinuity during the frequency change
between a logical 1 and 0 as seen by the waveform plot in Figure 3.30. The
MSK frequency spectrum decreases much faster than the QPSK spectrum, so
it is more efficient. Gaussian minimum shift keying (GMSK) first passes the
bits through a Gaussian filter before entering the MSK modulator. A Gaussian
filter has a smooth transfer function with a spectrum that falls off even faster
than the MSK spectrum and has no zero crossings. Its transfer function and
89
3 Passband Signals
1
0.5
VQPSK
90
0
−0.5
−1
0
10
20
30
40
t (ns)
50
60
70
80
Figure 3.29 Plot of a 𝜋/4 QPSK signal.
1 GHz 1.5 GHz
1
0
1 GHz
1
1 GHz
1
Figure 3.30 MSK modulation.
1.5 GHz
0
t (ns)
2
4
6
8
impulse response are given by
(
H(f ) = e
−
0.5887f
B
)2
(3.29)
3 −(5.3365Bt)2
(3.30)
e
B
In the time domain, it minimizes the rise and fall times and has no overshoot
to a step function input.
Differential phase shift keying (DPSK) uses relative phase rather than
absolute phase to represent symbols. Detecting a phase change is easier
than accurately determining the phase of the received signal, because the
demodulator does not need a coherent carrier. When DPSK transmits a “1,”
the phase changes by 180∘ while transmitting a “0” has no phase change.
Figure 3.31 shows an example of a DPSK signal. Note the phase changes at 4
and 6 ns.
h(t) =
3.5 Quadrature Amplitude Modulation
Quadrature amplitude modulation (QAM) combines ASK and PSK in order
to generate more distinguishable and longer symbols than either ASK or PSK
alone. A symbol change results in an amplitude and/or PM of the carrier. Points
3.5 Quadrature Amplitude Modulation
Figure 3.31 DPSK signal
example.
1
0
1
1
0
t (ns)
2
4
6
8
Q
Figure 3.32 IQ diagram for a 16-QAM
signal.
3V
1011
1001
0010
0011
0000
0001
1V
3V
1V
1010
1000
1101
1100
0100
0110
1111
1110
0101
0111
I
on the constellation diagram have radii that are proportional to the amplitude
of the modulated signal, while the angular location of the points correspond to
the phase of the modulated signal. As the signal changes symbols (moves from
one point to another on the constellation diagram), amplitude and PM occurs.
Figure 3.32 displays a constellation diagram for a 16-QAM signal. The symbol
0011 has the same phase as 0000 but a higher amplitude. In contrast, the symbol
1011 has the same amplitude as 0011 but a different phase.
Adding noise to the signal creates scatter plots about the constellation points
as shown in Figure 3.33. For instance, if the SNR = 16 dB (Eb/N0 = 10 dB),
then 16-QAM results in the constellation in Figure 3.33a with a BER = 0.002.
Adding more noise so that SNR reduces to 10 dB (Eb/N0 = 4 dB), results in a
greater spread in the scatter plots about the 16 constellation points as shown
in Figure 3.33b. Its BER increases to 0.0766. In addition, intersymbol interference (ISI) occurs because a point on the constellation diagram representing one
symbol moves into the detection region of another symbol. Grid lines separate
symbol detection regions. Points that lie within a 2 by 2 V box are interpreted as
the symbol in the center of the box (indicated by a “+”). The ISI in Figure 3.33b is
noticeably greater than in Figure 3.33a, because many points from one symbol
lie within the box of another symbol.
91
3 Passband Signals
6
6
4
4
2
2
Q (V)
Q (V)
92
0
0
−2
−2
−4
−4
−6
−6 −4 −2
0 2
I (V)
4
6
−6
−6 −4 −2
0 2
I (V)
4
6
Figure 3.33 Constellation diagrams for 16-QAM. (a) SNR = 16 dB and (b) SNR = 10 dB.
Q
Figure 3.34 Star 16-QAM
constellation.
1011
1111
1001
0011
0111
1101
0001
0000
0101
1000
I
0010
0100
0110
1010
1100
1110
QAM constellation points do not have to be in a rectangular grid as in
Figure 3.32. One variation, called star QAM [3] is PSK with two or more
amplitude levels (Figure 3.34). Star QAM minimizes error rates for a given
transmission power. The star 16-QAM symbols with the same amplitude in
Figure 3.32 are 45∘ apart in phase.
3.6 Power Spectral Density of Digital Signals
A baseband signal defined over an interval T has a normalized average power
of
T∕2
1
s2 (t)dt W
T→∞ T ∫−T∕2
Pavg = lim
(3.31)
3.6 Power Spectral Density of Digital Signals
Parseval’s theorem says that the integral of the square of a signal equals the
integral of the square of its transform (S(f ) = ℑ{s(t)}). Applying this theorem to
(3.31) produces
∞
1
|S(f )|2 df W
T→∞ T ∫−∞
The PSD is the integrand in (3.32)
Pavg = lim
(3.32)
|S(f )|2
W∕Hz
(3.33)
T→∞
T
Normalized average power in the frequency domain is the PSD integrated over
all frequencies.
PSDB (f ) = lim
∞
Pavg =
∫−∞
PSD(f )df W
(3.34)
Table 3.1 lists the baseband PSD expressions for several digital modulation
schemes. All the equations are in terms of T b even though most modulation
techniques deal with symbols where the symbol period is
Ts = Tb log2 M
(3.35)
The energy per bit equals the square of the carrier amplitude times the bit
period
Eb = A2c Tb
(3.36)
The PSD for the bandpass signal is written in terms of the baseband PSD as
]
1[
PSDP =
(3.37)
PSDB (f − fc ) + PSDB (f + fc )
4
Table 3.1 Baseband PSD for selected modulations. Reminder: sinc(x) = sin(𝜋x)/(𝜋x).
Modulation
ASK
M-ASK
M-FSK
PSDB
Eb
[𝛿(f )∕Tb + sinc2 (fTb )]
2
Eb
[𝛿(f )∕Tb + sinc2 (fTb )]
2
{
}
M
Eb
E
∑
sinc2 [(f − fm )Tb ] + b sinc2 [(f + fm )Tb ]
4M
4M
m=1
BPSK
Eb sinc2 (fT b )
QPSK
2Eb sinc2 (2fT b )
M-PSK
Eb log2 (M)sinc2 (nfT b log2 (M))
OQPSK
2Eb sinc2 (2fT b )
Eb
sinc2 (fTb log2 M)
2
M-QAM
93
94
3 Passband Signals
This formula allows the simple conversion of PSDB to a bandpass signal modulated by a carrier.
The spectral or bandwidth efficiency of a modulation scheme is the bit rate
divided by the bandwidth.
𝜂se =
Tb log2 M
Rb
= 0.5log2 M bits∕s∕Hz
=
B
2∕Tb
(3.38)
Spectral efficiency increases as M increases, but symbols get closer together
and require a higher SNR. Increasing the symbol energy increases the SNR.
3.7 BER of Digital Signals
Closed form expressions exist for the BER associated with some common digital modulation techniques [4]. ASK has the following BER:
(√ )
Eb
(3.39)
BER = Q
N0
BPSK, QPSK, and OQPSK have the same BER:
)
(√
2Eb
BER = Q
N0
where the Q and the complementary error functions are defined as
)
(
1
z
Q(z) = erfc √
2
2
2
erfc(z) = √
𝜋 ∫z
(3.40)
(3.41)
∞
2
e−𝜉 2 d𝜉
(3.42)
Other modulation schemes have more complicated expressions that apply
under special circumstances, so they are not listed here.
3.8 Multiplexing in Time and Frequency
A simplex wireless system conveys information one way – from a transmitter
to a receiver. Radio stations transmit music to user radios that receive it and
cannot transmit back. No communication occurs in the reverse direction.
In contrast, a duplex wireless system allows transmission and reception by
all devices on the network. A full duplex system allows users to simultaneously transmit and receive, like a telephone. Frequency division duplexing
(FDD) transmits at one frequency and receives at another frequency. Satellite
3.8 Multiplexing in Time and Frequency
communications use a C band uplink at 5.925–6.425 GHz and a downlink
at 3.7–4.2 GHz. A half-duplex system means that only one device transmits
at a time. In a half-duplex system like walkie-talkie radios, one person says
“over” when finishing a transmission in order to let the other person know
that the channel is open for transmission. Time division duplexing (TDD) has
designated times for transmit and receive signals.
Multiplexing combines multiple signals over a shared medium. The process
of multiplexing divides a communication channel into a number of subchannels with each message assigned to a different subchannel. Demultiplexing
extracts the multiplexed signals from the different subchannels. Frequency
division multiplexing (FDM) divides the spectrum into bands and assigns
signals to the bands. Time division multiplexing (TDM) divides time into slots
then assigns signals to time slots. This section introduces FDM and TDM
before explaining the more versatile multiple access schemes.
3.8.1
Frequency Division Multiplexing
FDM divides a specified frequency band into nonoverlapping subbands with
each subband having a carrier frequency. It places multiple independent signals in a designated spectrum (e.g. AM or FM radio) or splits a high-data-rate
signal into multiple lower-data-rate signals that exist in parallel. All the subbands combine to form a composite signal that occupies a designated portion
of the spectrum.
Figure 3.35 shows a diagram of an FDM system that has M signals modulated into subbands that occupy the composite signal bandwidth. Either each
subband has its own transmitter at a unique carrier frequency, or a single transmitter sends the entire composite signal. At the receiving end, the composite
BPF
Modulator
s1
Demodulator
s1
fc1
fc1
fc1
Composite signal Bandwidth
s2
fc2
Composite
signal
fc1
fc2
fcM
s1
s2
sM
s2
fc2
Composite
fc2
signal
Subband
bandwidth
sM
sM
fcM
fcM
Figure 3.35 Frequency division multiplexing with M subbands.
fcM
95
3 Passband Signals
signal passes through a bandpass filter (BPF) centered at the carrier frequency
of one subband. The filtered signal is then demodulated to recover the baseband signal. A good example of FDM is the television UHF Band from 470 to
806 MHz. Each station, numbered 14–69, has a 6 MHz slot. Radio astronomy
fills a gap at station 37, 608–614 MHz.
Orthogonal frequency division multiplexing (OFDM) divides a high-datarate signal into a number of low-data-rate signals then transmits them in
parallel by modulating with subcarriers spaced 1/T s apart in order to make
all the low-data-rate signal spectra orthogonal. An orthogonal subcarrier has
a peak in its spectrum when all the other subcarrier spectra equal zero in
order to reduce adjacent channel interference. The subcarrier bands do not
have to be contiguous. A conventional digital modulation scheme (such as
QPSK, 16-QAM, etc.) modulates the carrier with a low-symbol-rate signal.
The combined data rates of the subcarriers result in an overall data rate
similar to conventional single-carrier modulation schemes with an equivalent
bandwidth.
Figure 3.36 shows three symbols transmitted in time with guard bands
between symbols. The OFDM representation in the frequency domain is
the inverse fast Fourier transform (IFFT) of the symbol. Each IFFT bin in
the frequency domain corresponds to an orthogonal subcarrier spectrum.
A subcarrier signal occupies a null-to-null bandwidth equal to 2/T s . OFDM
usually has several channels with guard bands between channels, so it can
handle many data streams.
3.8.2
Time Division Multiplexing
TDM combines several low-bit-rate signals into one high-speed bit stream by
assigning signals to specific time slots. Synchronous TDM assigns each signal to
a predefined slot recognized by the receiver. All signals have the same sampling
fc
fc + 1/ Ts
fc + 2 /Ts
fc + 3 /Ts
96
Figure 3.36 Time–frequency
diagram for OFDM.
…
Guard
intervals
f
IFFT
t
Ts
FFT
Orthogonal
subcarriers
3.8 Multiplexing in Time and Frequency
Transmitter a
Transmitter b
Transmitter c
Mbps
8
16
Multiplexer
Receiver a
… a bb c c c a b bc c c …
24
Demultiplexer
Receiver b
Receiver c
Figure 3.37 Example of synchronous TDM.
rate, so all data streams have the same bit rate which allows the transmitter and
the receiver to perfectly synchronize with the slot period. If the signal data rate
goes down, then slots go unused and efficiency drops.
Example
Given three signals with bit rates of 8, 16, and 24 Mbps, use TDM to combine
them into one channel.
Solution
The TDM channel has a transmission rate of 48 Mbps (sum of the bit rates of the
three signals). The ratio of the three data rates is 8 : 16 : 24. Thus, the time slots
are allocated in proportion to 1 : 2 : 3. The ratio sums to 6, which corresponds
to the minimum length of the slot assignment period. The slot assignment is
illustrated in Figure 3.37.
Statistical TDM does not have synchronized time slots with fixed assignments for each different data stream. Instead, the system dynamically assigns
time slots based on the past, current, and predicted data rates of the different
signals [5]. Allocating time slots based on need allows more users to participate. This approach promises to be more efficient but requires some intelligent
processing.
3.8.3
Multiple Access
In TDM/FDM, known users have assigned time slots/subbands. Multiplexing
works well for a fixed number of users. In a wireless system where users come
and go, like a cellular network, dynamic assignment using multiple access techniques proves to be much more efficient.
The multiple access approach in packet radio, allows several transmitters to
send bursts of data packets to a receiver at any time. Each packet competes for
the receiver’s attention. In an early research project on radio access to computer
systems, Professor Norman Abramson of the University of Hawaii invented a
simple multiple access approach called ALOHA in which many remote terminals share one radio channel without central control [6]. The receiver responds
with an acknowledgement once it successfully receives a packet. If two packets
97
98
3 Passband Signals
Transmitter
1
Successfull
Collision
2
3
Figure 3.38 ALOHA
packets as a function of
time. If packets overlap,
then they are
retransmitted.
4
5
6
7
8
t
from different transmitters overlap, then a collision occurs and the packets are
not received. If the transmitter does not get an acknowledgement after a short
time, then it resends the packet. Figure 3.38 shows an example where eight
transmitters randomly send packets. Packets that do not overlap are successfully received. Transmitters must resend packets that collide.
Assume that the users of an ALOHA system generate random packets at a
rate that follows a Poisson process [7] in which 𝜆p packets that are 𝜏 p long are
transmitted per unit of time. The probability of no collisions means that no
other transmission occurs over the period t − 𝜏 p to t + 𝜏 p . The probability of no
collisions equals the probability that no other packets transmit over the interval
t − 𝜏 p to t + 𝜏 p and is given by
p(no collisions) = e−2𝜆p 𝜏p
(3.43)
The ratio of the average rate of successfully transmitted packets (𝜆p ) divided
by the channel packet rate and is a unitless quantity called throughput:
𝛾p = 𝜆p 𝜏p e−2𝜆p 𝜏p
(3.44)
ALOHA’s throughput increases when a user checks the channel availability
before transmitting. Another improvement to ALOHA forces all packets into
established time slots rather than random time slots. Slotted ALOHA reduces
collisions due to small overlaps of packets. Collisions still occur when two or
more packets occupy a time slot as shown in Figure 3.39. Collisions in slotted
ALOHA only occur over the interval t to t + 𝜏 p , so the probability of no packet
collisions in an interval is given by [8]
p(no collisions) = e−𝜆p 𝜏p
(3.45)
with a corresponding throughput of
𝛾p = 𝜆p 𝜏p e−𝜆p 𝜏p
(3.46)
Figure 3.40 shows a plot of the throughput as a function of load (𝜆p 𝜏 p ) for
ALOHA and slotted ALOHA. A small load means that few users transmit
packets over the channel. As the load increases, more transmitters send
3.8 Multiplexing in Time and Frequency
Figure 3.39 Slotted
ALOHA.
Transmitter
Successfull
Collision
1
2
3
4
5
6
7
8
Figure 3.40 Throughput as a
function of load for ALOHA and
slotted ALOHA.
t
0.4
γp
0.3
Slotted
ALOHA
0.2
ALOHA
0.1
0
0
1
3
2
4
5
λ pτ p
packets that result in more collisions. ALOHA has a peak throughput of 0.184
when 𝜆p 𝜏 p = 0.5 which means that successful transmission only occurs 18.4%
of the time. Slotted ALOHA improves throughput with a peak of 0.368 when
𝜆p 𝜏 p = 1.0 which means that successful transmission occurs 36.8% of the time.
In either case, these approaches are extremely inefficient for transmitting data
over a wireless network. An Aloha network needs some central management,
or it becomes overloaded and the throughput falls to zero. Although novel at
the time, more efficient approaches to multiple access are currently available.
Frequency division multiple access (FDMA) is FDM that allows users access
to any open subband in the channel. Orthogonal frequency-division multiple
access (OFDMA), a multi-user version of OFDM, dynamically assigns subcarriers to users based on the availability of subbands. Figure 3.41 illustrates the
difference between OFDM and OFDMA. OFDM assigns all the subcarriers to
a user in a given time slot whereas OFDMA assigns subcarriers in a given time
slot to multiple users. Some OFDMA time slots have no users assigned to subcarriers due to lack of demand.
Time division multiple access (TDMA) shares a single-carrier frequency with
several users where users have nonoverlapping time slots. TDMA data transmission occurs in bursts and has low battery consumption, because the transmitter only turns on when transmitting. A user gets a time slot only when
actively transmitting.
99
3 Passband Signals
f
OFDM
Subcarriers
100
OFDMA
1
2
3
4
1
2
2
1
2
3
4
1
2
2
1
2
3
4
1
2
1
2
3
4
4
2
3
2
1
2
3
4
4
1
3
2
1
2
3
4
4
1
3
1
2
3
4
4
3
1
2
3
4
4
3
4
3
4
3
3
4
3
4
3
3
4
3
1
1
1
1
User 1 data
2
User 2 data
4
3
User 3 data
4
4
User 4 data
No data
t
Time slots
Figure 3.41 OFDMA spectrum vs. OFDM slot assignment.
3.9 Spread Spectrum
Spread spectrum forces a signal with bandwidth B to occupy a larger bandwidth of BGp , where the processing gain Gp > 1.0. The extra bandwidth spent
on spread spectrum overcomes several problems encountered in the channel.
Spread spectrum originated during World War II, when the actress
Hedy Lamarr learned how easily the enemy jammed and steered away
radio-controlled torpedoes [9]. She originated the idea of hopping the torpedo
control signals from one frequency to another to avoid jamming. In order to do
so, the frequency hopping requires synchronization between the transmitter
and the receiver on the torpedo. Her pianist friend, George Antheil, helped
her develop a synchronization process based on the idea of a miniaturized
player-piano. Their patent for frequency-hopping spread spectrum [10] was
too difficult to implement at the time, so the first practical application did not
occur until 1962 on Navy ships during the Cuban missile crisis. Since then,
the military developed many different spread-spectrum techniques which
eventually became common in the commercial sector (e.g. Bluetooth – see
Appendix IV).
The channel-capacity theorem illustrates the tradeoff made when converting
a narrowband signal into a wideband signal. For a constant channel capacity,
decreasing the SNR requires increasing the bandwidth. If C = 10 Mbs and
SNR = 20 dB, then B = 66 MHz. To maintain the same capacity, if the SNR
decreased to −10 dB, then B must increase to 3.2 GHz. Thus, a high-power narrowband signal spreads into a low-power wideband signal while maintaining
the same channel capacity. Burying the signal into the noise is very attractive for
military or spy applications. It allows spectrum sharing if the signal just appears
3.9 Spread Spectrum
Transmit
Receive
Transmit
d
Receive
Interference
d
Power
Power
Interference
Signal
Signal
Minimum
SNR
Minimum
SNR
Noise
Noise
dmax
(a)
d
dmax
d
(b)
Figure 3.42 Signal, interference, and noise power levels in a channel. (a) A minimum SNR
limits dmax when only the signal and noise are present and (b) a minimum SINR limits dmax
when the signal, noise, and interference are present.
to be noise. Spread-spectrum transmit power levels are similar to narrowband
signal power levels but have a much lower spectral power density. This means
that spread spectrum and narrowband communication systems will not interfere with each other when operating in the same frequency band.
3.9.1
Interference
Power decreases as the separation distance between the transmitter and
receiver increases. Noise remains relatively constant, though, so at a maximum
distance, dmax , the SNR drops below the minimum level for reliable communication. Figure 3.42a shows a plot of the signal power decay vs. distance from
the transmitter as well as the constant noise floor or receiver sensitivity. The
signal power level that yields the minimum SNR dictates dmax as indicated.
An interfering signal also experiences a drop in power with distance from the
receiver as shown in Figure 3.42b. The interference reduces dmax , because its
power exceeds that of the noise. A minimum SINR then determines a reduced
dmax rather than a minimum SNR.
3.9.2
Frequency-Hopping Spread Spectrum
Frequency hopping requires synchronization between the receiver and the
transmitter. The hopping sequence comes from a pseudo random noise (PRN)
code that has a spectrum similar to a random sequence of bits. The PRN code
determines the order of the frequencies transmitted. The receiver decodes
101
102
3 Passband Signals
f
01
11
00
11
11
01
10
00
Symbol
(a)
4-FSK
(b)
SFH
(c)
FFH
t
Ts
Figure 3.43 Frequency-hopping spread spectrum. (a) 4-FSK signal, (b) 4-FSK signal with
SFH, and (c) 4-FSK signal with FFH.
the signal by following the same PRN sequence as the transmitter. In order to
initialize the synchronization process, the frequency hopper starts at a fixed
frequency before hopping to other predetermined frequencies. As long as the
jammer does not interfere with the start frequency, the frequency hopping
occurs over a much broader bandwidth than the jamming signal, so the desired
signal usually avoids the jamming. The bandwidth of a frequency-hopped
signal equals the number of available frequency slots times the bandwidth of a
slot. Certain bands can be avoided if they contain potential interference.
Slow frequency hopping (SFH) means that at least one symbol transmits in
every frequency subband. Fast frequency hopping (FFH) means that more than
one frequency hop occurs during a symbol. SFH can use coherent detection,
while FFH cannot, so SFH is more commonly used. Figure 3.43a has the symbols for a 4-FSK. If this FSK signal slow hops between four frequency channels
at the rate of one change in frequency per symbol then Figure 3.43b shows one
possible hopping sequence. If the fast hopping occurs twice in one symbol, then
Figure 3.43c shows a possible hopping sequence. In both cases, the frequency
hopping bandwidth is four times that of the original signal.
An FM signal with a 100 kHz bandwidth that hops over a 100 MHz bandwidth
has 1000 possible subbands. For equally likely subbands, the spread-spectrum
signal interferes with any narrowband receiver that has a 100-kHz bandwidth
at one of the hops only 1/1000 of the time. As a result, the average interference
power is 1000 times less than the transmitted power. This low level of interference does not interfere with an FM radio as long as the hopping is fast enough.
Frequency hopping within the high bandwidth of a television signal might cause
the loss of one of the picture lines that viewers would notice, so avoid frequency
hopping in the television spectrum.
3.9 Spread Spectrum
Data pulse
t
Figure 3.44 Time hopping a data pulse within established time slots.
Time-hopping spread-spectrum positions a pulsed RF carrier within a time
interval in accordance with a PRN code sequence (Figure 3.44) [11]. Time hopping combined with frequency hopping creates a TDMA spread-spectrum system.
3.9.3
Direct-Sequence Spread Spectrum
In order to spread the spectrum of a narrowband signal, a PRN sequence first
modulates the code word as shown in Figure 3.45. One bit in the PRN sequence,
called a chip, has a period of T cp . The modulated data stream now has a bandwidth increase equal to the spreading factor: F sp = T b /T cp . After spreading, the
data modulates a carrier frequency then transmits to the receiver.
Figure 3.45 illustrates the interference rejection capability of direct-sequence
spread spectrum (DSSS). A narrowband high-power density interference mixes
with the low-power density spread-spectrum signal in the channel. The receiver
down converts the signal and interference to baseband using the same PRN
sequence as the transmitter. The down conversion de-spreads the baseband
signal but spreads the interference. In order for the de-spreading to work, the
PRN synchronizes with the baseband spread-spectrum signal. The synchronizer locks onto the strongest baseband multipath signal in order to align its
chips with the PRN. A matched filter or correlator identifies the bits at the
receiver in Figure 3.45. Interference power spreads over a bandwidth that is F sp
times bigger than the original interference bandwidth. Since the power density
PRN
PRN
LO
cos(2πfct)
∼
Input
×
×
LO
cos(2πfct)
Channel
∼
Synchronizer
×
×
Output
∫
Signal
Signal
f
Interference
Signal
f
Interference
f
f
Figure 3.45 Narrowband interference in the channel has little impact on the received signal
in a DSSS system.
103
104
3 Passband Signals
f
f
TDMA
t
f
FDMA
Figure 3.46 TDMA and
FDMA lie in the
time–frequency plane.
CDMA adds another
dimension called code.
CDMA
t
t
Code
of the interference decreases by F sp , it has little impact on the desired signal.
The processing gain is the increase in signal power density after de-spreading.
3.9.4
Code Division Multiple Access (CDMA)
Code division multiple access (CDMA) is a multiple user version of DSSS. Each
user has a different PRN code, so that a receiver has access to one signal in a
channel when many signals occupy the same bandwidth. Figure 3.46 illustrates
the difference between TDMA, FDMA, and CDMA. TDMA signals exist in
time slots while FDMA signals exist in frequency bands. CDMA signals, on
the other hand, fill up the time–frequency plane but are separated in a third
dimension called code.
CDMA comes in two forms: synchronous and asynchronous. Synchronous
CDMA or code division multiplexing (CDM) requires all signals in the channel
to precisely align in time. This works with point to multi-point communication,
such as transmission from a base station to mobile users. Synchronous CDMA
uses orthogonal codes (e.g. Walsh codes) to spread the signals. Asynchronous
CDMA, on the other hand, allows signals to arrive randomly as with communication between a mobile unit and a base station. No orthogonal codes exist
that spread signals with arbitrary starting points. The orthogonal codes used in
synchronous CDMA do not correlate well with asynchronous CDMA signals,
so PRN codes or Gold codes [10] are used instead.
The Walsh codes used in CDM come from rows of Hadamard matrices. The
elements of a Hadamard matrix are either 1 or −1. Its rows are mutually orthogonal and Hn HTn = 2n I2n . A Hadamard matrix results from a recursive relationship that starts with the initial matrices:
]
[
1 1
(3.47)
H0 = [1] and H1 =
1 −1
then creates larger square matrices using the formula
]
[
Hn Hn
Hn+1 =
for a 2n+1 × 2n+1 Hadamard matrix
Hn −Hn
(3.48)
3.9 Spread Spectrum
Example
Generate H 2 and H 3 using MATLAB.
Solution
The MATLAB commands are hadamard(4) and hadamard(8) where the
argument is the number of rows/columns.
⎡1 1 1 1 1 1 1 1 ⎤
⎢
⎥
⎢1 −1 1 −1 1 −1 1 −1⎥
⎢
⎥
⎡1 1 1 1 ⎤
⎢1 1 −1 −1 1 1 −1 −1⎥
⎢1 −1 −1 1 1 −1 −1 1 ⎥
⎢1 −1 1 −1⎥
⎥ H3 = ⎢
⎥
H2 = ⎢
⎢1 1 1 1 −1 −1 −1 −1⎥
⎢1 1 −1 −1⎥
⎢
⎢
⎥
⎥
⎣1 −1 −1 1 ⎦
⎢1 −1 1 −1 −1 1 −1 1 ⎥
⎢
⎥
⎢1 1 −1 −1 −1 −1 1 1 ⎥
⎢1 −1 −1 1 −1 1 1 −1⎥
⎣
⎦
Asynchronous CDMA, uses a PRN code or Gold code to spread the signal.
For instance, GPS signals use Gold codes. These codes provide a better
correlation than orthogonal codes when the bits are not perfectly aligned.
On the other hand, the asynchronous codes raise the noise floor, because the
cross-correlation of the codes is not zero like it is with the orthogonal codes.
Example
Three receivers each have different 4-bit PRN codes:
PRN code A = [1 − 1 − 1 − 1]
[
]
PRN code B = −1 −1 1 1
[
]
PRN code C = 1 1 −1 1 .
A transmitter sends the message 1 0 0 1 using PRN A to encode it. Show that
a receiver using PRN A will receive the message while a receiver using PRN B
or C will not.
Solution
Assuming that a 1 bit is encoded by “1” and a 0 bit by “−1,” then the encoded
message sent by the transmitter is given by
[
] [
]
1 −1 −1 1 = 1 −1 −1 −1 −1 1 1 1 −1 1 1 1 1 −1 −1 −1
The message and codes are graphed in Figure 3.47. The receiver using PRN
A takes
the dot ]product of the encoded sequence with A, 4 bits at a time to
[
4
−4
get
take the dot product with B to yield
[
] −4 4 . The other[ two receivers
]
2 2 2 2 and C to yield 0 0 0 0 . Note that decoding with B produces a
105
Bits
1
0.5
0
Message
1
0
−1
A
1
0
−1
B
3 Passband Signals
1
0
−1
C
106
1
0
−1
0
500
1000
1500
0
500
1000
1500
0
500
1000
1500
0
500
1000
1500
0
500
1000
1500
Figure 3.47 Plots of the bits in the message, the signal, and the codes for the three
transmitters.
nonzero
output.
If these codes were orthogonal, then this dot product would
[
]
be 0 0 0 0 . In this case, the dot product with B appears to be noise.
Problems
3.1
If V c = 10 V and V m = 5 V, find:
(a) The modulation coefficient
(b) The Percent modulation.
3.2
If an AM signal has a peak of 20 V and a minimum positive amplitude of
6 V, determine:
(a) The modulation coefficient
(b) The carrier amplitude.
3.3
For an envelope with a maximum of 30 V and a positive minimum of
10 V, determine:
(a) The unmodulated carrier amplitude
(b) The peak change in the amplitude of the envelope
(c) The modulation coefficient
(d) The percent modulation.
3.4
Describe the following expression for an amplitude-modulated wave in
terms of frequency content and voltage amplitude:
s(t) = 6 sin 2𝜋300 × 103 t − 2 cos 2𝜋312 × 103 t + 2 cos 2𝜋288 × 103 t
Problems
3.5
For V c = 12 V and 𝛽 AM = 0.5, determine the following:
(a) The percent modulation
(b) The peak voltages of the carrier and the side frequencies
(c) The maximum amplitude of the envelope, V max
(d) The minimum amplitude of the envelope, V min
(e) Plot the AM envelope (label all pertinent voltages).
3.6
For a modulation coefficient 𝛽 AM = 0.4 and a peak carrier power Pc =
400 W, determine:
(a) The peak power in each sideband
(b) The total sideband power
(c) The total transmitted power.
3.7
Use MATLAB to amplitude modulate a 300 kHz carrier with a voice signal. Use your own voice file or the one provided. Plot the time domain
signal when 𝛽 AM = 0.33, 0.67, 1.0, 1.33.
3.8
An ASK signal transmits data at 28.8 kbps over a channel with a bandwidth from 300to 3400 Hz.
(a) How many symbol states are needed?
(b) How many symbol states are needed if the channel passband goes
from 0 to 3400 Hz and baseband signaling was used.
(c) What is the theoretical capacity if the SNR = 33 dB?
3.9
Determine the percent modulation for a television broadcast station
with Δf = 50 kHz when f m = 30 kHz.
3.10
Determine the number of side frequencies produced for the following
FM modulation indices: 0.25, 0.5, 1.0, 2.0, 5.0, and 10.
3.11
For an FM modulator with 𝛽 FM = 5, modulating signal = 2 sin(27𝜋5 ×
103 t), and a carrier at 400 kHz, determine the following:
(a) The number of sets of significant sidebands
(b) The sideband amplitudes.
3.12
For an FM transmitter with 80 kHz carrier swing, determine the frequency deviation. If the amplitude of the modulating signal decreases
by a factor of 4, determine the new frequency deviation.
3.13
A modulated signal takes the form s(t) = 100 cos[2𝜋(10 × 106 )t + 4 sin
2𝜋(103 )t]
(a) Find the modulation index and bandwidth if the signal is FM.
(b) Repeat (a) if f m is doubled.
107
108
3 Passband Signals
(c) Find the modulation index and bandwidth if the signal is PM.
(d) Repeat (c) if f m is doubled.
3.14
Use MATLAB to frequency modulate a 300 kHz carrier with a voice signal. Use your own voice file. Plot the time domain signal when 𝛽 FM = 0.67,
1.33, 2.67.
3.15
What happens to the demodulated DSB signal when the receiver LO has
a frequency of f 1 = f c + 𝛿f instead of f c ?
3.16
Create 10 000 I and Q bits. Plot points in I–Q plane for (a) SNR = 6 dB,
(b) SNR = 10 dB, and (c) SNR = 20 dB.
3.17
A video signal with a bandwidth from 0 to 2 MHz is sampled at four
times the highest frequency using a 16 bit ADC. This data is modulated
to 16-QAM with a raised cosine filter having 𝛼 = 0.5. What is the transmitted video signal bandwidth?
3.18
Random data is BPSK modulated and sent at 4800 bps over a bandpass
channel. Find the transmission bandwidth such that the frequency spectrum is below 35 dB outside of the band.
3.19
An hour of temperature data sampled at 50 kHz and quantized to 16 bits
is stored in memory. How many bits are stored in memory?
3.20
Plot the normalized analytical PSD in dB/Hz for (a) ASK, (b) BPSK, (c)
QPSK, (d) 4-QAM, and (e) 16-QAM when T b = 1 μs.
3.21
Plot the BER for QPSK vs. Eb/N0.
3.22
A source uses the same average power to transmit ASK and PSK. Does
ASK or PSK have a faster data rate for a given bit error probability at the
receiver.
3.23
Plot the throughput as a function of load for ALOHA and slotted ALOHOA.
3.24
A DSSS system has four codes based on the 4 × 4 Hadamard matrix. Generate 4, 16 random bit sequences where a one is represented by “+1” and
a zero is represented by a “−1.” Spread the bit sequence n using row n
of the Hadamard matrix. The output is a 4 × 16 = 64 bit sequence. Add
the four spread messages together into 1 total bit sequence. Show row
References
n of the Hadamard matrix recovers bit sequence n from the total bit
sequence.
3.25
Repeat Problem 21 but add noise to the bit sequences. Can the bit
sequences be recovered when there is a low SNR?
3.26
GPS uses Gold codes for spreading the signal. Use MATLAB to demonstrate transmission and reception of a binary signal using Gold codes.
References
1 Carson, J.R. (1963). Notes on the theory of modulation. Proceedings of the
IEEE 51 (6): 893–896.
2 http://www.rfwireless-world.com/Terminology/QPSK-vs-OQPSK-vs-pi-
4QPSK.html (accessed 18 December 2018).
3 Hanzo, L.L., Ng, S.X., Keller, T., and Webb, W. (2004). Star QAM Schemes
for Rayleigh Fading Channels, 307–335. Wiley-IEEE Press.
4 https://www.mathworks.com/help/comm/ug/bit-error-rate-ber.html
(accessed 30 July 2019).
5 Win, M.Z. and Scholtz, R.A. (2000). Ultra -wide bandwidth time-hopping
6
7
8
9
10
11
spread-spectrum impulse radio for wireless multiple-access communications. IEEE Transactions on Communications 48: 679–691.
Abramson, N. (1970). The ALOHA system – another alternative for computer communications. Proc. Fall Joint Computer Conf., AFIPS Press 37:
281–285.
Gold, R. (1967). Optimal binary sequences for spread spectrum multiplexing (Corresp.). IEEE Transactions on Information Theory 13 (4): 619–621.
Goldsmith, A. (2013). Wireless Communications. New York: Cambridge
University Press.
https://www.youtube.com/watch?v=NI8nOa9BvjY (accessed 27 June 2018).
Markey, H.K. and Antheil, G. (1942). Secret communication system. US
Patent 2, 292,387, issued 11 August 1942.
Molisch, A.F., Zhang, J., and Miyake, M. (2003). Time hopping and frequency hopping in ultrawideband systems. In: 2003 IEEE Pacific Rim
Conference on Communications Computers and Signal Processing (PACRIM
2003) (Cat. No.03CH37490), vol. 2, 541–544. Victoria, BC, Canada: IEEE.
109
111
4
Antennas
Signals move charges. When charges accelerate, they radiate. A charge moving
on a curved path or bouncing off the end of a wire accelerates/decelerates and
creates a time-changing electromagnetic field that propagates away at the speed
of light. Antennas confine charges to accelerate in ways that cause signals to
radiate in desirable directions.
Transmit antennas radiate a modulated signal into one end of a channel that
travels to the receive antenna at the other end of the channel. The antenna’s
ability to receive or transmit a signal depends on the antenna size, pointing
direction, polarization, and frequency response. This chapter introduces basic
antenna design principles for a wireless system.
4.1 Signal Properties that Influence Antenna Design
The transmitter and receiver along with the channel characteristics influence
the antenna design specifications. As explained in Chapter 2, the signal has a
direction, a spectrum, and a polarization. The antenna in a wireless system must
have enough gain to increase the signal power, operate in the same spectrum
as the signal, and match the signal’s polarization.
4.1.1
Impedance
Like any circuit element, an antenna has an impedance represented by a
complex number written as
Zant = Rr + RL + jX a
(4.1)
The resistive loss (RL ) due to lossy material in the antenna converts some of the
wireless energy into heat. Radiation resistance (Rr ) and antenna reactance (X a ),
on the other hand, result from the antenna geometry and material composition.
Thus, any signal arriving at the transmit antenna terminals reflects, converts to
heat, or radiates. The impedance usually determines the antenna’s operating
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
112
4 Antennas
bandwidth. An antenna’s impedance should match that of the feedline over the
operating bandwidth in order to insure maximum power transfer between the
transmitter and the antenna or the antenna and the receiver.
4.1.2
Gain
An antenna is a spatial filter that concentrates a signal in a specific direction
as shown in Figure 4.1. This spatial filter response, the antenna pattern, has a
main beam which is the spatial passband over a defined angular region, as well
as sidelobes and nulls due to diffraction in the stopband.
An isotropic point source radiates equally in all directions from a single point
in space (dashed line in Figure 4.1). It serves as a standard for comparing antennas that magnify the signal in some directions over others. Directivity describes
an antenna’s ability to concentrate power in one direction more than other
directions. The directivity of any antenna equals the maximum radiated power
density divided by the average radiated power density which equals the power
density radiated by an isotropic point source:
D=
4𝜋
2𝜋
∫0
𝜋
∫0
|AP(𝜃, 𝜙)|2 sin 𝜃d𝜃d𝜙
(4.2)
where AP(𝜃, 𝜙) is the normalized antenna pattern (peak value of 1).
The antenna gain combines directivity with antenna dissipative losses. Gain
is defined as the radiation intensity in a given direction divided by the radiation intensity of an isotropic point source accepting the same power. Gain does
not include losses arising from impedance and polarization mismatches and is
independent of its wireless system. Gain relates to the directivity by
G(𝜃, 𝜙) = 𝛿e D(𝜃, 𝜙)
(4.3)
where 𝛿 e is the radiation efficiency (ratio of the power radiated by the antenna
to the power input to the antenna). Realized gain, yet another descriptor of the
antenna radiation, includes mismatch loss between the feed and the antenna.
The term "gain" without any angular dependence, G, means the maximum gain
of the antenna and is usually expressed in decibels (10 log G in dB).
G
Isotropic point source pattern
Null
Main beam
Sidelobe
Null
Sidelobe
Antenna
Figure 4.1 The antenna pattern is
the response as a function of angle.
4.1 Signal Properties that Influence Antenna Design
The effective (or equivalent) isotropic radiated power (EIRP) of a transmitter
multiplies the gain of a transmitting antenna (Gt ) by the net power delivered to
the transmitting antenna (Pt ).
EIRP = Pt Gt
(4.4)
EIRP describes how effectively a transmitter concentrates power in a given
direction.
The effective area (Ae ) of a receiving antenna describes its ability to collect
radio frequency (RF) signals. Effective area is the ratio of the available power at
the receiving antenna output to the power flux density of a plane wave incident
on the antenna. It relates to the gain of the receiving antenna by
4𝜋Ae
(4.5)
𝜆2
For many antennas, the effective aperture is proportional to the physical
aperture (Ap ):
G=
Ae = 𝛿a Ap
(4.6)
where 𝛿 a is an aperture efficiency and 0 ≤ 𝛿 a ≤ 1.0.
Example
Find the directivity of an antenna with AP(𝜃) = cos 𝜃 in the upper half plane
(𝜃 ≥ 0) and zero in the lower half plane.
Solution
Substitute into (4.2) to get
D=
4𝜋
2𝜋
∫0
𝜋
∫0
cos 𝜃 sin 𝜃d𝜃d𝜙
=
4𝜋
=4
𝜋
or 6 dB.
4.1.3
Polarization
Polarization defines the direction of the electric field vector as a function of
time. If a plane wave travels in the z-direction, the electric field lies in the x–y
plane and takes the form:
⃗ = Ex0 cos(𝜔t − kz)̂
x + Ey0 cos(𝜔t − kz + Ψy )̂
y
E(t)
(4.7)
This electric field vector traces an ellipse in the x–y plane over one period.
Figure 4.2 shows the electric field rotation for left hand and right hand elliptical polarization. Pointing the right thumb in the direction of wave propagation,
fingers curl in the direction of the E field trajectory when the wave is right hand
113
114
4 Antennas
→
E (t)
Left hand
polarized
→
E (t)
Direc
t
propa ion of
gation
Right hand
polarized
Figure 4.2 Rotation of the electric field for right-hand and left-hand polarization.
polarized (RHP) or 180 ∘ < Ψy < 360∘ . Otherwise, it is left hand polarized (LHP)
or 0 ∘ < Ψy < 180∘ .
Axial ratio (AR) describes the shape of the polarization ellipse of an antenna
where
AR =
Length of major axis
Length of minor axis
(4.8)
An AR of 1 (0 dB) indicates circular polarization. Linear polarization has an
AR = ∞ because the minor axis is zero, and the amplitude of the peak of the
electric field vector lies along a straight line. Circular polarization occurs when
x and ̂
y components are 90∘ out of phase. An antenna has the
Ex0 = Ey0 , and the ̂
same polarization, hence AR, whether it transmits or receives.
A receive antenna that does not have the same polarization as the incoming
electromagnetic wave will not receive the maximum possible power. Polarization loss factor (PLF) accounts for the polarization mismatch between an
incident wave and an antenna’s polarization and is given by
𝛿p = |̂ei ⋅ ̂e∗r |2
(4.9)
where
̂ei = polarization vector of incident wave =
Ei
|Ei |
̂er = polarization vector of receive antenna =
Ei = incident electric field
Eantenna = electric field of antenna.
Eantenna
|Eantenna |
PLF indicates the reduction in the received power due to the polarization mismatch. A receive antenna with the same polarization as the transmit antenna
receives all the power when the two antennas directly face each other. Antenna
polarization changes with angle off boresight.
4.1 Signal Properties that Influence Antenna Design
Example
What is the polarization of the following fields?
Ex
1
0.707
0.707
0.867
4.1.4
Ey
0
0.707
0.707
0.5
Ψy
45∘
0
90∘
90∘
Solution ∶
x linear
linear 45∘ from x-axis
LHP circular
elliptical
Bandwidth
Antennas serve as bandpass frequency filters that operate best from a low
frequency (f lo ) to a high frequency (f hi ). Outside this range of frequencies,
the antenna performs at a degraded level that does not meet acceptable
standards. The values of f lo and f hi depend on one or more of the following
specifications:
1. Antenna gain: The upper and lower frequency limits define the region where
the antenna gain stays above −3 dB or half of the peak gain at the center
frequency.
2. SWR: The upper and lower frequency limits define the region where the
SWR is less than 2 or the reflection coefficient <−10 dB. In other words,
at least 90% of the power goes to the antenna.
3. Polarization: A circularly polarized antenna has a 0 dB AR at the center frequency. The high and low frequencies where the AR increases to 3 dB marks
the polarization bandwidth.
A wideband antenna has a bandwidth greater than 10% otherwise, the
antenna is narrow band. In 1990, a Defense Advanced Research Projects
Agency (DARPA) Ultra-Wideband Radar Review Panel defined ultra-wide
band (UWB) as any system where B ≥ 25% when f hi and f lo are 20 dB below
the peak power density [1]. Later in 2002, the Federal Communications
Commission (FCC) defined UWB as B ≥ 25% or B ≥ 15 GHz [2]. These two
government agencies used different definitions for f hi and f lo .
Example
An amplitude modulation (AM) antenna operates between f lo = 540 and
f hi = 1600 kHz. What is its bandwidth?
Solution
f c = 1070 kHz B = f hi − f lo = 1060 kHz, B =
99.065% and B =
fhi
flo
= 1600∕540 = 2.963.
fhi −flo
fc
× 100 = 1060∕1070 × 100 =
115
116
4 Antennas
4.2 Common Antennas
Wireless systems require many types of antennas with a wide range of capabilities that have tradeoffs depending upon the application. This section categorizes antennas as either point sources, wire antennas, aperture antennas, or
microstrip antennas. These arbitrary groupings cover most antennas of importance in wireless communications.
4.2.1
Point Sources
An antenna appears to radiate from a single point when observed at a great
distance. For example a star looks like a point source of light on earth even
though it is physically huge. The phase center of an antenna looks like a point
where the radiation originates. It lies at the center of an imaginary sphere of
equal phase radiating from the antenna. Point sources serve as approximations
for modeling the phase interactions of multiple antennas.
An antenna in Figure 4.3 transmits a spherical wave front from a single point.
The wavefront propagates to receive antenna 1 at a distance R1 then on to
receive antenna 2 at a distance R2 . The phase difference between the center
and edge of antenna 1 is greater than at antenna 2 (ΔR1 > ΔR2 ). According to
the Institute of Electrical and Electronics Engineers (IEEE), the antenna far field
starts when the separation distance is
R = 2D2max ∕𝜆
(4.10)
At this distance, the incident wave approximates a plane wave (ΔR ≤ 𝜆/16 ⇒
kΔR ≈ 0) [3].
Example
Derive (4.10) given the geometry in Figure 4.3.
Constant amplitude
and phase
R1
Transmit antenna
+
1
R2 +
R1
R2
ΔR
Receive
antenna 1
ΔR 2
Dmax
Receive
antenna 2
Figure 4.3 Receive antenna in the near field and the far field.
4.2 Common Antennas
Solution
2
2
(Dmax / 2) + R = (R + ΔR)
2
2
0
D2max / 4 + R2 = R2 + 2Rλ /16 + (λ /16)
R=
2D2max
λ
4.2.2
Wire Antennas
A twin-wire transmission line (left side of Figure 4.4) has current flowing in
opposite directions on the two wires. The currents on these wires have electric
fields that point in the same direction between the wires but in opposite directions everywhere else. Consequently, the fields outside the wires cancel but the
fields between the wires add and propagate. Bending the ends of a two-wired
transmission line by 90∘ , as shown in right side of Figure 4.4, forces the currents
on the bent ends to flow in the same direction along the z-axis. Now, the fields
radiated by the bent wires radiate. This simple antenna is called a dipole.
The most common type of dipole is half of a wavelength long (𝓁 = 𝜆/2). It has
a standing wave at the resonant frequency, reminiscent of a vibrating string. If
the dipole lies along the z-axis and has a current I dipole flowing in the wires, then
the electric field at a distance r in the far field is [4]
)
(
k𝓁
cos
𝜃
− cos(k𝓁∕2)
−jkr cos
2
e
E𝜃 = jZ0 Idipole
(4.11)
2𝜋r
sin 𝜃
Observation
point
y
r
Transmission line
Electric
field lines
l
Bend
z
wires
θ
ϕ
Current
x
Figure 4.4 Bending the ends of a transmission line creates a dipole that radiates an electric
field.
117
118
4 Antennas
z
z
θ
y
y
x
(a)
(b)
Figure 4.5 Magnitude of the electric field of a half-wavelength dipole. (a) 3D pattern and (b)
cut of 3D pattern in the y–z plane showing variation in 𝜃.
where Z 0 = 377Ω = characteristic impedance of free space. Its antenna pattern
is the dipole response at one frequency as a function of angle. The normalized
dipole antenna pattern (AP) has a peak of one.
)
(
cos
𝜃
− cos(k𝓁∕2)
cos k𝓁
2
AP(𝜃) =
(4.12)
sin 𝜃
Since the dipole current only flows in the z-direction, the electric field is z- or
𝜃-polarized depending upon whether the coordinate system is rectangular or
spherical. AP has a maximum at 𝜃 = 90∘ , and nulls arise when AP = 0. Nulls
in the AP occur at 𝜃 = 0∘ and 𝜃 = 180∘ . Since the dipole is symmetric in the
𝜙 direction, (4.12) is a function of 𝜃 and independent of 𝜙 as shown by the 3D
antenna pattern and the pattern cut in Figure 4.5.
Half wavelength dipoles have a 73Ω input impedance at resonance. A
feed line having an impedance of 73 Ω insures maximum power transfer. An
impedance match insures maximum power transfer when the feed line and
antenna impedances differ. Dipoles made from thin wires have a very narrow
impedance bandwidth, while fat wire versions have a much wider bandwidth.
Example
Find the directivity of a half-wavelength (𝓁 = 𝜆/2) dipole. Assume the power
density equals the magnitude of the electric field squared.
Solution
Substitute (4.12) into (4.2) to calculate the directivity. Since Srmax is in the
numerator and denominator, all the constants divide out leaving
D=
2𝜋
∫0
𝜋
∫0
[
(
cos
4𝜋
𝜋
2
cos 𝜃
sin 𝜃
= 1.643 = 2.16 dB
) ]2
sin 𝜃 d𝜃 d𝜙
(4.13)
4.2 Common Antennas
Inner conductor
Antenna pattern
+
λ/4
=
Ground plane
Coax
Figure 4.6 Making a monopole from a coaxial cable and a circular ground plane with a hole
in the middle.
Extending a coaxial cable center conductor 𝜆/4 through a hole in a ground
plane (Figure 4.6) and soldering the outer conductor to the ground plane
forms a simple monopole. A monopole has half the input impedance of a half
wavelength dipole (Zin = 36.5 + j21.25 Ω) but twice the directivity, since it only
radiates in the upper hemisphere (0 ≤ 𝜙 < 360∘ and 0 ≤ 𝜃 < 90∘ ). A monopole
antenna pattern looks like half of a donut and is omnidirectional in azimuth (𝜙).
Practical monopoles rarely look like the ideal one in Figure 4.6. The monopole
on top of the car in Figure 4.7a is not perpendicular to the ground plane in
order to make it more aerodynamic. Common radio monopoles rarely have
any noticeable ground plane like the one in Figure 4.7b. Broadcast monopoles
(Figure 4.7c) have a wire ground screen buried a few centimeters below the
surface that serves as the ground plane. The impedance of a nonideal monopole
differs from the theoretical perfect monopole. A receive antenna tolerates an
impedance mismatch better than a transmit antenna, because high-power
reflections from the transmit antenna may harm sensitive electronics.
The folded dipole (Figure 4.8) joins the two ends of a dipole to form a
squashed loop with dimensions 𝓁 = 𝜆/2 long and d ≪ 𝓁. Its 292 Ω impedance
(a)
(b)
(C)
Figure 4.7 Examples of monopole antennas. (a) Car, (b) AM/FM radio, and (c) HF
communications.
119
4 Antennas
Figure 4.8 Folded dipole in Tekapo, New Zealand.
ℓ
d
rs
ed
tiv
s
ctor
Dire
Figure 4.9 An HF Yagi–Uda antenna
with 1 active dipole, 3 reflectors, and
12 directors.
ipo
le
cto
Refle
Ac
120
Director
Folded dipole
Figure 4.10 Folded dipole has high
impedance that is significantly
lowered by the presence of parasitic
elements.
Reflector
nearly matches a 300-Ω transmission lines [5]. Folding the dipole increases its
bandwidth.
A Yagi–Uda antenna has one active dipole that lies in the same plane as many
parasitic (passive) dipoles of different lengths (Figure 4.9). This design increases
the gain and bandwidth of a single dipole antenna [6]. The parasitic elements
reduce the impedance of the active element, so using a high impedance dipole,
like the folded dipole, for the active element counters this impedance reduction (Figure 4.10). Passive elements longer than the resonant length have an
inductive impedance (current lags voltage) that reflects a signal. Directors are
4.2 Common Antennas
shorter, capacitive (current leads voltage), passive dipoles. By placing one or
two reflectors on one side of the active element and array directors on the other
side, the phase distribution across the elements points the beam away from the
reflectors and toward the directors.
A log periodic dipole array (LPDA) looks similar to a Yagi [7] but has all its elements connected to the feed (no passive elements). The half wavelength dipoles
have resonant frequencies, lengths, diameters, and spacings that are logarithmically proportional. Elements much shorter than a half wavelength are too
capacitive to radiate, while elements much longer than a half-wavelength are
too inductive to radiate, so nearly all of the radiation comes from the resonant
element and the two adjacent elements. Figure 4.11 shows a vertically polarized HF (high frequency) log periodic monopole array made from monopoles
instead of dipoles. An LPDA operates over a very wide bandwidth (Yagi is narrow band) but has a lower gain compared to the Yagi.
Ham radio operators invent ingenious alternatives to the very tall monopole
at HF frequencies where a wavelength is between 10 and 100 m. For instance,
the inverted L antenna bends a monopole by 90∘ so that most of the wire runs
parallel to the ground (Figure 4.12a). Ham radio operators like the inverted
High frequency
LPDA
Low frequency
Figure 4.11 HF log periodic monopole array.
L
(a)
sp
L
h
(b)
h
w
(c)
Figure 4.12 Diagrams of (a) inverted L antenna, (b) inverted F antenna, and (c) PIFA
121
122
4 Antennas
L antenna, because trees and buildings support the long antenna instead of a
tall tower. Shorting the arm to the ground plane at a distance sp from the feed
point decreases the length of the L antenna at the resonant frequency (keeps
it out of your neighbor’s yard). Now, this inverted F antenna looks like the letter F rotated 90∘ (Figure 4.12b). Decreasing sp reduces the input impedance
[8, 9]. The planar inverted-F antenna (PIFA) (Figure 4.12c) is a popular handset
antenna made from a small L × w metal patch. The feed is at the center of the w
dimension and along the L-dimension, while the shorting post is placed along
the w-dimension. A rule of thumb for the PIFA resonant frequency is [10]
c
(4.14)
f0 =
4(w + L)
Some PIFA designs perform well in multiple frequency bands [11]. Figure 4.13
shows a patented dual band PIFA. The high- and low-band antennas have their
arms wound to reduce their length, so that they fit inside a handset.
A small wire loop antenna has a circumference less than one-tenth of a wavelength. Current around a small loop is approximately constant. The magnetic
field close to a small loop dominates the electric field, so it is often called a magnetic field probe in the near field. A small loop has very low input impedance
that requires a match to a feed line such as a coaxial cable. This large mismatch
makes the small loop an adequate receive antenna but a poor transmit antenna.
Thus, the small loop is typically used as a receive antenna and not a transmit
antenna (too much power reflected back to a transmitter).
Low band
34
High band
30
20
5
40
32
24
50
A2
42
G2
22
A1
G1
Shorts
Feeds
Figure 4.13 Diagram of a two band PIFA [12].
10
4.2 Common Antennas
y
Figure 4.14 Diagram of a loop antenna in
the x–y plane.
Loop antenna
rl
x
ϕ
θ
Antenna pattern
z
The small loop antenna in Figure 4.14 has electromagnetic fields given
by [4]
Er = E𝜃 = H𝜙 = 0
[
]
Z0 (k0 r𝓁 )2 I0 sin 𝜃
1
E𝜙 =
1+
e−jk0 r
4r
jk 0 r
Hr = j
k0 r𝓁2 I0 cos 𝜃
H𝜃 = −
2r2
[
]
1
1+
e−jk0 r
jk 0 r
]
[
(k0 r𝓁 )2 I0 sin 𝜃
1
1
e−jk0 r
−
1+
4r
jk 0 r (k0 r)2
(4.15)
In the far field, the 1/r2 and 1/r3 terms quickly become very small a short distance from the loop, so only the E𝜙 and H 𝜃 terms remain. A loop is linearly
polarized in the x–y plane and has a directivity of 1.5. Its electric field goes to
zero at 𝜃 = 0∘ , so the antenna pattern has a null perpendicular to the plane
containing the loop as seen by the antenna pattern in Figure 4.14.
Adding multiple turns to the loop increases its input impedance in order
to better match to the feed line. If the small loop of radius r𝓁 has N turn turns
(Figure 4.15a), then its radiation resistance depends on the number of turns
squared.
( r )4
𝓁
2
(4.16)
Rr = 31.171𝜋 2 Nturn
𝜆
Adding turns increases the loop impedance until it matches its feed line. AM
radio receivers often use multi-turn loop antennas with a ferrite core to increase
the radiation efficiency (Figure 4.15b).
123
124
4 Antennas
Figure 4.15 Multi-turn loops (a)
antenna with Nturn = 4 and (b) AM
radio multi-turn loop with ferrite
core.
Multi-turn loop
Feed
Ferrite core
(a)
(b)
Example
Design a multi-turn loop antenna that has a radiation resistance of 50 Ω at 800 MHz. Assume the
loop has a radius of 3 cm and the wavelength is
37.5 cm.
Solution
Substitute the given data into (4.16) to get
)
(
3 4
2
⇒ Nturn = 63
50 = 31.171𝜋 2 Nturn
37.5
z
Axial mode
pattern
θ
Normal mode
pattern
dsp
rl
A helical antenna extends the center conductor
of a coaxial cable then twists it into the shape of a Figure 4.16 The two modes
cork screw of radius r𝓁 with spacing dsp as shown of a helical antenna.
in Figure 4.16. Unlike the multi-turn loop, only one
end of the helical antenna connects to the feed. If the total wire length is much
less than a wavelength, then it behaves like a monopole and operates in the normal mode. The normal mode helix has a maximum gain perpendicular to the
axis of the helix (like a monopole) and is linearly polarized with the electric field
parallel to its axis. A normal mode helix connected to a monopole produces an
antenna length shorter than 𝜆/4 as shown in Figure 4.17.
An axial mode helix (dsp ≈ 𝜆/4 and r𝓁 ≈ 1/k) has circular polarization with a
peak gain along its axis. The AR at 𝜃 = 0∘ for an axial mode helix with N hel turns
is approximately [13]
AR =
Monopole
2Nhel + 1
2Nhel
(4.17)
Helix
Figure 4.17 Monopole shortened by
a helix [14].
4.2 Common Antennas
An approximate formula for the directivity axial mode helix is [13]
( )
)
(
dsp
2𝜋r𝓁 2
Nhel
D = 12
𝜆
𝜆
(4.18)
As 𝜃 increases, the directivity decreases, and the antenna becomes elliptically
polarized. The impedance of the helical antenna depends upon its feed. One
approximate formula for an axial mode helical antenna radiation resistance
with an axial feed is [13]
r
(4.19)
Rr = 280𝜋 𝓁
𝜆
Example
Design an axial mode helical antenna that has circular polarization at 3 GHz
with directivity of 6 dB and a radiation resistance of 50 Ω.
Solution
The wavelength at 3 GHz is 10 cm.
r
Rr = 280𝜋 𝓁 = 50 ⇒ r𝓁 = 0.5684 cm
10
Substituting into (4.18) and solving for the number of turns results in
( )
)
(
dsp
2𝜋 × 0.5684 2
D = 4 = 12
Nhel
⇒ Nhel dsp = 26.1343
10
10
Assume that dsp = 3 cm then
Nhel = 9
4.2.3
Aperture Antennas
An aperture refers to an opening that radiates/collects electromagnetic radiation. An a × b rectangular slot with a uniform electric field has a relative diffraction pattern at a point (x0 , y0 ) in a plane that is z0 away from the aperture is given
by [15]
)
(
)
(
by0
ax0
sin c
(4.20)
AP ≃ sinc
𝜆z0
𝜆z0
Figure 4.18 shows the relative antenna pattern of a rectangular aperture when
a > b. As a/𝜆 and b/𝜆 get large, their respective sinc pattern main beams get
narrower. Its diffraction pattern is the Fourier transform of a uniform electric
field in the rectangular aperture. The first sidelobe is 13.26 dB below the main
beam, and the first null is at 𝜃 null1 = 57.3 ∘ /(a/𝜆). The point on the main beam
that is 3 dB below the peak is at 𝜃 3dB = 50.8 ∘ /(a/𝜆), so the 3 dB beamwidth in
the x–z plane is 𝜃 3dB = 101.6 ∘ /(a/𝜆) and in the y–z plane is 𝜃 3dB = 101.6 ∘ /(b/𝜆).
125
126
4 Antennas
Figure 4.18 Rectangular aperture
and its diffraction pattern.
y
x
(x0,y0)
z
a
b
z0
A circular aperture with radius ra and a uniform√electric field has the following relative diffraction pattern at a distance r0 =
from the aperture [15]:
(
)
kr r
J1 za 0
0
AP ≃
kr r
x20 + y20 from a point z = z0
(4.21)
a 0
z0
Its first sidelobe is 17.6 dB below the main beam while the first null is at
𝜃 null = 35 ∘ /(a/𝜆). The 3 dB beamwidth is 𝜃 3dB = 29.2 ∘ /(ra /𝜆). In reality,
the fields in the apertures have a polarization and an amplitude and phase
variation (mode), so their antenna patterns are more complicated [4]. The
expressions in (4.20) and (4.21) take the form of a typical antenna pattern by
converting the variables to spherical coordinates. The directivity depends on
the aperture area: Ap = ab for a rectangular aperture and Ap = 𝜋ra2 for a circular
aperture.
D≈
4𝜋Ap
𝜆2
(4.22)
Example
Assume that a 12 cm × 6 cm rectangular aperture with a 10 GHz uniform field
projects onto a flat screen in the far field. Plot the relative diffraction pattern
over a 100 cm by 100 cm area that is centered on the aperture.
4.2 Common Antennas
Solution
The MATLAB code is given below and the resulting pattern is shown in
Figure 4.19.
f=10e9; lam=3e10/f;
a=12; b=6;
z0= 2*(aˆ2+bˆ2)/lam;
x0=-100:.5:100; y0=x0;
[xx,yy]=meshgrid(x0,y0);
AP=sinc(a*xx/(lam*z0)).*sinc(b*yy/(lam*z0));
figure(1);mesh(x0,y0,abs(AP))
xlabel(' x (cm)');ylabel('y (cm)');zlabel('|AP|')
A leaky feeder communication system uses a coaxial cable with holes or apertures in the outer conductor to emit and receive radio waves, thereby functioning as an extended antenna (Figure 4.20) [16]. The system has a limited range
and its operating frequency (typically VHF or UHF [VHF, very high frequency;
UHF, ultra high frequency]) cannot pass through dirt and rock. The leaky feeder
Figure 4.19 Radiation from a
rectangular aperture.
⏐AP⏐
1
0.5
0
100
100
0
y (cm)
0
−100
−100
x (cm)
Transport mode
RF in
RF out
Radiating mode
Figure 4.20 Leaky coaxial cable antenna [17]. Source: courtesy of Centers for Disease
Control and Prevention.
127
128
4 Antennas
finds applications in mines, underground railways, commercial airplanes, and
buildings for communications and Wi-Fi. Line amplifiers inserted at regular
intervals (350–500 m) keep the signal strength above a set threshold level [17].
The leaky cable interfaces with portable transceivers for two-way radio communication.
4.2.4
Microstrip Antennas
A microstrip antenna, also known as a patch or printed antenna, is an area
of copper etched from or routed on (usually just one side) a printed circuit
board (PCB). The metal on the bottom of the PCB serves as a ground plane.
The advantages of microstrip antennas include:
•
•
•
•
•
Low fabrication cost
Conforms to curved surfaces
Easy to group elements into a large array
Light weight
Easily integrates with electronics.
Disadvantages include:
•
•
•
•
•
Low power handling
Narrow bandwidth
Low efficiency
Surface waves in the substrate cause unwanted coupling
Radiation from feeds and junctions.
Some modifications to the patch counter the disadvantages and make
microstrip antennas useful for many applications.
Microstrip patches come in a variety of shapes with the most common being a
L × w rectangle. At 10 GHz, the rectangular microstrip patch has dimensions of
about 1 cm × 1 cm, so it easily fits onto a PCB. On the other hand, at 100 MHz,
the 1 m × 1 m patch does not fit on a PCB. The PCB size limitation means that
patch antennas rarely operate at frequencies below 900 MHz. The dominant
patch polarization leaks out of the radiating edges as shown by the arrows in
Figure 4.21. The fields at these edges are in phase. In contrast, the field lines
along the long edges switch phase at the halfway point. Consequently, the fields
cancel rather than radiate. Any radiation coming out of the long edges is orthogonal to the dominant polarization, so it is called cross-polarized.
Electric field lines under the patch resonate with maxima at the radiating
edges and zero in the center of the patch. The ratio of the electric to the magnetic field at the point where the transmission line feeds the patch determines
the input impedance. The edge-fed microstrip line or bottom-fed coaxial cable
attaches to the patch at a point that corresponds to 50 Ω. An alternative to the
bottom-fed coax is a stripline or an aperture feed underneath the patch.
4.2 Common Antennas
Figure 4.21 Diagram of a
rectangular microstrip
patch antenna.
Radiating
edges
Microstrip
line feed
L
w
εr
h
Coaxial
cable feed
Patch designs typically start by approximating the length and width of the
patch using design formulas. Two simple formulas are given by [18]
𝜆
𝜆
(4.23)
L ≃ 0.49 √0 , w ≃ √ 0
𝜀r
2(𝜀r + 1)
where 𝜀r is the relative permittivity of the PCB substrate. These dimensions
serve as seeds for a numerical optimization algorithm in an antenna design
software package [20]. Substrate height and permittivity primarily determine
patch bandwidth [19].
h
B∝ √
𝜀r
(4.24)
A typical patch has a bandwidth of a few percent. The two normalized electric
field polarizations associated with rectangular patches are [5]
(
)
(
)
w
L
sin 𝜃 sin 𝜙 cos
sin 𝜃 cos 𝜙 cos 𝜙
E𝜃 = sin c
𝜆(
) 𝜆(
)
w
L
E𝜙 = − sin c
sin 𝜃 sin 𝜙 cos
sin 𝜃 cos 𝜙 cos 𝜃 sin 𝜙
(4.25)
𝜆
𝜆
The total electric field is
√
|Et (𝜃, 𝜙)| = E𝜃2 + E𝜙2
(4.26)
Figure 4.22 shows a plot of (4.26) with principal plane cuts. The principal planes
pass through the peak gain and are at 𝜙 = 0∘ and 𝜙 = 90∘ . When 𝜙 = 0∘ then
E𝜙 = 0, so the field is 𝜃-polarized. When 𝜙 = 90∘ then E𝜃 = 0, so the field is
𝜙-polarized. For linearly polarized antennas, the E-plane contains the electric
field, while the H-plane contains the magnetic field.
Circularly polarized patches have two different modes excited in the cavity
under the patch with one phase delayed by 90∘ relative to the other [21]. A
simple feed approach uses a Wilkinson power divider that feeds two different
locations on the patch. The power divider splits the signal evenly. One of the
signals receives a 90∘ phase shift (longer path). Other approaches include feeding the patch along the diagonal and mitering the patch corners. As with other
antennas, the AR changes with frequency and angle off boresight.
129
4 Antennas
ϕ = 90° cut
ϕ
=
0°
cu
t
1
z
y
x
E (V/m)
130
0.5
0
−90
−45
0
45
90
θ (deg)
Figure 4.22 Total electric field of patch antenna with orthogonal principal plane cuts.
Example
Estimate the dimensions of a rectangular patch on a substrate with 𝜀r = 2.2 that
operates at 2.5 GHz.
Solution
Use (4.23) to estimate the patch dimensions: 𝜆0 =
12
= 3.96 cm,
L ≃ 0.49 √
2.2
3×1010
2.5×109
= 12 cm
12
= 4.74 cm
w= √
2(2.2 + 1)
4.3 Antenna Arrays
An antenna array combines many smaller antennas, called elements, into a
larger antenna. The array size and hence its gain increases as N (number of
elements) increases. Phased arrays have phase shifters that electronically steer
the main beam in a desired direction. Electronic beam scanning surpasses slow
mechanical antenna steering that tends to frequently breakdown. Appropriately, weighting the signals at the elements before adding them together results
in low sidelobes or moving nulls in order to reject unwanted signals. Arrays
also gracefully fail. In other words, if one antenna element fails, the array still
functions but at a lower performance level. Remote locations like outer space
and on top of tall towers where repairs are extremely difficult benefit from
an array’s robust performance. The many performance advantages of a phased
array antenna come at a high cost proportional to the number array elements.
4.3 Antenna Arrays
z
Figure 4.23 Array geometry relative with an incident
plane wave.
C
on
a
st
ph
ds
in
θ
nt
e
as
θ
θ
1
d
2
x
Figure 4.23 shows a plane wave (constant amplitude and phase) incident at
an angle of 𝜃 on an array with two elements separated by d. The plane wave first
arrives at element 2 then at element 1 with a time delay of
Δ = (d∕c) sin 𝜃
(4.27)
The transmitted signal st (t) arrives at both elements and adds together to get
the received signal.
sr (t) = st (t) + st (t − Δ)
(4.28)
At a single frequency, f , the time difference converts to phase by
𝜓 = 2𝜋f Δ = 2𝜋f (d∕c) sin 𝜃 = kd sin 𝜃
(4.29)
The array factor equals the sum of the signals at one frequency:
AF = 1 + ej𝜓
(4.30)
AF depends on f , d, and 𝜃. The transmitting array phase has a 180∘ phase difference from the receive array, so its array factor is given by
AF = 1 + e−j𝜓
(4.31)
The phase sign indicates whether the plane wave travels toward (positive) or
away (negative) from the array.
4.3.1
Element Placement
The variables Δ and 𝜓 depend on the element locations. When N signals add
together in phase (𝜓 = 0∘ ), then the resulting signal amplitude increases by N.
For example if 𝜃 = 0∘ , then sr (t) = 2st (t) and AF = 2. The main beam or array
factor maximum points at 𝜃 = 0∘ . For all other angles, AF(𝜃) ≤ AF(0).
4.3.1.1
Linear Array
A signal arriving at an N element array with element n located at xn (Figure 4.24)
encounters the closest element first then continues to successive elements until
it reaches the last one. Each element receives the signal at a different time. The
131
4 Antennas
z
Figure 4.24 Diagram of an N element linear array with
a plane wave incident at 𝜃.
θ
(n
−1
)d
sin
θ
132
1
Pl
an
e
wa
d
ve
w1 w2
n
wn
N
x
wN
Σ
weighted (wn ) and time delayed (𝜏 n ) sum of N element signals produces the
output signal [22]:
sr (t) =
(
)
x
wn st t − n sin 𝜃 + 𝜏n
c
n=1
N
∑
(4.32)
An array factor represents the weighted and phase shifted sum of one frequency
or tone from each element.
AF =
N
∑
wn ej(kxn sin 𝜃 + 𝛿n ) =
n=1
N
∑
wn ej𝜓n
(4.33)
n=1
A phased array has phase shifters that change the signal phase at element n
by 𝛿 n .
A uniform linear array has wn = 1 and elements spaced d apart along the
x-axis: xn = (n − 1)d. The main beam peak points at broadside (𝜃 = 0∘ ). Placing
the phase center (𝜓 = 0∘ ) at the first element of the array results in an array
factor given by
AF = 1 + ej𝜓 + ej2𝜓 + · · · + ej(n−1)𝜓 =
N
∑
ej(n−1)𝜓
(4.34)
n=1
Multiplying both sides of (4.34) by ej𝜓 and subtracting the resulting product
from (4.34) leads to a simpler expression for the array factor
AF =
sin(N𝜓∕2) j N−1 𝜓
1 − ejN𝜓
=
e 2
j𝜓
sin(𝜓∕2)
1−e
(4.35)
The maximum of (4.35) occurs when 𝜓 = 0∘ .
AF =
N
∑
ej(n−1)0 = N
(4.36)
n=1
Dividing (4.35) by N normalizes the array factor to a main beam peak of 1.0.
Moving the phase center to the physical center of the array causes the phase
4.3 Antenna Arrays
term in (4.35) to disappear and leaves the normalized array factor:
AFN =
sin(N𝜓∕2)
N sin(𝜓∕2)
(4.37)
AFN has a first sidelobe approximately 13 dB below the main beam peak.
Example
Plot the magnitude of the unnormalized array factors for N = 4, 6, and 8 element
arrays with elements spaced d = 𝜆/2 along the x-axis.
Solution
Figure 4.25 has a plot of the array factors. The eight-element array has the highest main beam peak, narrowest main beam, and most sidelobes.
The beamwidth determines the antenna resolution and relates to the directivity of the array. A uniform linear array with half wavelength spacing has an
approximate 3 dB beamwidth of [23]
𝜃3dB ≃ 101.5∘ ∕N
(4.38)
Increasing the number of elements decreases the beamwidth. A half wavelength
spaced linear array has a directivity given by
|2
|∑N
| n=1 wn |
|
|
D = ∑N
|w |2
n=1 | n |
(4.39)
which for a uniform linear array with d = 𝜆/2 is
(4.40)
D=N
Adding more elements results in a higher directivity and a skinnier main beam.
Figure 4.25 Array factors
for N = 4, 6, and 8 element
arrays along the x-axis
with d = 𝜆/2.
8
AF
6
4
2
−90
0
θ (deg)
90
133
134
4 Antennas
Example
Find the 3 dB beamwidth for a six-element array with d = 𝜆/2 and compare to
(4.38).
Solution
√
Find the 3-dB beamwidth by setting (4.37) equal to 1∕ 2 and solving for 𝜃 using
the MATLAB command fzero:
sin(3𝜋 sin 𝜃)
1
= √ ⇒ 𝜃 = 8.6∘ ⇒ 𝜃3dB = 17.2∘
6 sin(0.5𝜋 sin 𝜃)
2
which is close to (4.38): 𝜃 3dB ≃ 101.5 ∘ /6 = 16.9∘
The approximation in (4.38) improves as N gets larger.
4.3.1.2
Arbitrary Array Layouts
Elements located anywhere in three-dimensional space (xn , yn , zn ) have an array
output represented by
sr (t) =
[
]
1
wn st t − (xn sin 𝜃 cos 𝜙 + yn sin 𝜃 sin 𝜙 + zn cos 𝜃) + 𝜏n
c
n=1
N
∑
(4.41)
Converting the time difference between elements to a phase at one frequency
results in an array factor given by
AF(𝜃, 𝜙) =
N
∑
n=1
wn ej[k(xn sin 𝜃 cos 𝜙 + yn sin 𝜃 sin 𝜙 + zn cos 𝜃) + 𝛿n ] =
N
∑
wn ej𝜓n
(4.42)
n=1
Planar arrays have all the elements in a plane (e.g. zn = 0). Figure 4.26 shows
a 4 × 4 uniform planar array of microstrip patches. Microstrip lines from
any element to the feed point at the center of the array have the same path
length, so each element has the same phase. A planar array with uniform
phase at the elements has its main beam pointing orthogonal to the plane
containing the elements. Note that the microstrip line width changes in order
Figure 4.26 A 4 × 4 uniform planar array of
microstrip patches.
Element
Feed
Match
4.3 Antenna Arrays
to match the impedance when the line splits and when it enters the patch.
Conformal arrays mold the elements to a structure, like an airplane fuselage.
Other configurations, such as spherical and random arrays, find specialized
uses in wireless systems.
Example
A planar array has N x elements spaced dx in the x-direction and N y elements
spaced dy in the y-direction arranged in a square grid for a total of N = N x × N y
elements. Use (4.37) to write an expression for the normalized array factor.
Solution
From superposition, the array factor is a product of the linear array factor from
the elements in the x-direction and the linear array factor in the y-direction.
(
)
)
(
Ny kdy sin 𝜃
Nx kdx sin 𝜃
sin
sin
2
2
AF =
(4.43)
(
)
(
)
kdy sin 𝜃
kdx sin 𝜃
Nx sin
Ny sin
2
2
A linear array with N elements has a 3 dB beamwidth of
)
(
0.443𝜆
rad
𝜃3dB = sin−1
Nd
(4.44)
The beamwidth of a planar array is defined by principal plane cuts at 𝜙 = 0∘ and
𝜙 = 90∘ .
At broadside, the directivity is proportional to the projected area of the array
(Ap ) as long as the element spacing is not much larger than 𝜆/2
D=
4𝜋Ap
𝜆2
𝜂t
where 𝜂 t is the taper efficiency defined by
(∑
)2
N
w
n=1 n
𝜂t =
∑N
N n=1 w2n
(4.45)
(4.46)
This formula assumes that the array only radiates in a hemisphere (e.g.
0 ≤ 𝜃 ≤ 90 ∘ , 0 ≤ 𝜙 < 360∘ ). If the array has an irregular shape, then assume that
each element occupies a unit cell that equals the area of dx dy for a rectangular
grid. Thus, an N element array has an approximate directivity given by
D = 𝜂t Ndx dy ∕𝜆2
(4.47)
135
4 Antennas
4.4 Electronic Beam Steering
The main beam peak of an array factor occurs when the phase term in (4.33) at
each element equals zero (𝜓 n = 0).
N
] ∑
[
AFmax = AF(𝜓n = 0∘ ) = AF (n − 1)kd sin 𝜃 + 𝛿n = 0∘ =
wn
(4.48)
n=1
The 𝛿 n term in 𝜓 n corresponds to the phase shifter setting at element n. Phase
shifters change the phase of a signal by 0∘ to 360∘ in increments defined by the
number of bits controlling them.
By selecting 𝛿 n such that 𝜓 n = 0∘ , the main beam points in the desired scan
direction, 𝜃 s :
𝛿n = −kxn sin 𝜃s
(4.49)
An array with the main beam pointing at 𝜃 s = 90∘ is called an end-fire array.
The value of 𝛿 n in (4.49) depends upon frequency (k = 2𝜋f /c), so if the frequency changes, the pointing direction of the main beam also changes. When
the array scans its beam, then the directivity decreases due to the decrease in
the projected area of the array.
D(𝜃s ) = D cos 𝜃s
(4.50)
Example
An eight-element uniform array with d = 𝜆/2 scans to 45∘ . Plot the magnitude
of the broadside and scanned array factors.
Solution
The scanning phase at element n is 𝛿 n = − 0.707𝜋(n − 1) radians. Figure 4.27
shows the resulting magnitude of the array factors calculated using (4.33).
Figure 4.27 The beam of
an eight element array
steered to 45∘ .
8
6
AF
136
θs = 0°
θs = 45°
4
2
−90
−45
0
θ (deg)
45
90
4.5 Element Pattern
4.5 Element Pattern
If a linear or planar array has elements like dipoles or microstrip patches rather
than isotropic point sources, then the array antenna pattern equals the element
antenna pattern times the array factor.
AP = Element pattern × Array factor
(4.51)
An array has the same polarization as its elements, so AP has vector components unlike AF.
Example
Assume an eight-element uniform array along the z-axis with d = 𝜆/2 has a sin𝜃
element pattern that is polarized in the z-direction. Plot the element pattern
superimposed over the array pattern (in dB) for the array factor scanned to
broadside and 45∘ .
Solution
Substitute the sin𝜃 element pattern into (4.42):
AP(𝜃) = sin 𝜃
8
∑
ejkzn (cos 𝜃−cos 𝜃s )
n=1
Figure 4.28 shows the array factor at broadside and steered to 45∘ . When
steered to broadside, the element pattern produces lower relative sidelobe
levels than the array factor. When the beam scans, the sidelobes become
asymmetric with respect to the main beam. As a result, the main beam peak
does not point in the desired direction, and the directivity decreases. Note that
the steered beam in Figure 4.27 does not show a decrease in directivity.
θs = 45°
Directivity (dB)
5
θs = 0°
Element
pattern
0
−5
−10
−15
−20
0
45
90
θ (deg)
135
180
Figure 4.28 Element pattern times the array factor equals the array pattern. As a result, the
array directivity decreases with scan.
137
138
4 Antennas
4.6 Low Sidelobes
Low sidelobes decrease signal amplitudes received through the sidelobes.
Transmit antenna low sidelobes attenuate signal propagation in directions
outside of the main beam in order to mitigate interference with other wireless systems in the same band. Amplitude weights at the elements produce
low sidelobe tapers. Several analytical formulas for amplitude tapers create
predictable low sidelobes in the array factor.
The z-transform plays a fundamental role in synthesizing low sidelobe amplitude tapers for arrays. A z-transform converts the array factor into a polynomial
by substituting z = ej𝜓 = ejkd sin 𝜃 into (4.34) to get [24]
AF =
N
∑
wn z(n−1) = w1 + w2 z + · · · + wN zN−1 = (z − z1 )(z − z2 ) · · · (z − zN−1 )
n=1
(4.52)
The zeros of this polynomial (zn ) correspond to the nulls in the array factor.
zn = ej𝜓n = ejkd sin 𝜃n
(4.53)
Remember that 𝜓 n represents the phase of the nth zero in AF. The actual null
in the array factor occurs at an elevation angle of 𝜃 n . The nth zero relates to null
n in the array factor by 𝜓 n = kd sin 𝜃 n . Moving one of these zeros results in a
new array factor polynomial that has the same order but different coefficients,
hence different element weights.
Table 4.1 summarizes the three most well-known low sidelobe amplitude
tapers for linear arrays, and the locations of the array polynomial zeros. Use
the following steps to find the low sidelobe amplitude weights and the corresponding array factor:
Table 4.1 Low sidelobe amplitude tapers [23].
Peak sidelobe level (dB)
𝝍n
Binomial [25]
−∞
180∘
Dolph–Chebyshev [26]
slldB
Taylor [27]
slldB
(
)
⎧
(n−0.5) 𝜋 ⎫
cos
⎪
⎪
N−1
2cos−1 ⎨
(
) ⎬
⎪ cosh 𝜋A
⎪
N−1
⎩
⎭
±2𝜋
N
√
⎧
A2 + (n − 0.5)2
⎪n
n<n
A2 + (n − 0.5)2
⎨
⎪
n
n≥n
⎩
A = 𝜋1 cosh−1 (10slldB∕20 ) and n − 1 is the number of zeros moved
4.6 Low Sidelobes
1.
2.
3.
4.
5.
Calculate 𝜓 n using Table 4.1.
Let zn = ej𝜓n .
Substitute zn into (4.52).
Multiply the factored polynomial to find the wn .
Calculate the array factor using (4.33).
Nulls in the array factor occur at
(𝜓 )
n
𝜃n = sin−1
kd
(4.54)
Example
Calculate the amplitude weights for a binomial, a 20 dB Chebyshev, and a 20 dB,
n = 2 Taylor taper, then plot the weights and associated array factors. Assume
the array has eight elements spaced 𝜆/2 apart.
Solution
Follow these steps:
Calculate 𝜓 n using Table 4.1. See Table 4.2 for values.
Let zn = ej𝜓n .
Substitute zn into (4.52).
Multiply the factored polynomial to get the polynomial coefficients or array
weights shown in Table 4.2 and Figure 4.29.
Taking these weights and calculating AF for 𝜙 = 0∘ and 0 ≤ 𝜃 ≤ 90∘ results in the
1.
2.
3.
4.
array factor plots in Figure 4.30. In order to emphasize the relationship between
𝜃 and 𝜓, 𝜓 axis appears below the 𝜃 axis, where
( )
2𝜋
0.5𝜆 sin 𝜃 = 𝜋 sin 𝜃 = 180∘ sin 𝜃
𝜓=
𝜆
Table 4.2 Location of zeros and elements weights for the four amplitude tapers in the
example.
Uniform
n
𝝍n
1
2
Binomial
Chebyshev
wn
𝝍n
0.79
1
3.14
0.03
0.94
0.58
0.95
0.55
1.57
1
3.14
0.20
1.55
0.66
1.57
0.68
3
2.36
1
3.14
0.60
2.33
0.88
2.36
0.87
4
3.14
1
3.14
1
3.14
1
3.14
1
wn
𝝍n
Taylor
wn
𝝍n
wn
5
−2.36
1
3.14
1
−2.33
1
−2.36
1
6
−1.57
1
3.14
0.60
−1.55
0.88
−1.57
0.87
7
−0.79
1
3.14
0.20
−0.94
0.66
−0.95
8
1
0.03
0.58
0.68
0.55
139
4 Antennas
1
Uniform
Amplitude
0.8
0.6
Taylor
0.4
Binominal
0.2
Chebyshev
0
1
20
Element
Figure 4.29 Low sidelobe tapers for an eight-element linear array.
10
Directivity (dB)
140
Binomial
0
Chebyshev
Taylor
Uniform
−10
−20
−30
−80
−180°
−60
−40
−20
0
20
θ (deg)
40
60
0°
80
ψ
180°
Figure 4.30 Array factors corresponding to the weights in Figure 4.29. There are 𝜃 and 𝜓
axes for comparison.
The array factors are normalized to the peak of the uniform array factor
(N = 8). Note that the array factor directivity decreases as the sidelobe level
decreases, so there is a tradeoff between increasing the desired signal entering
the main beam and decreasing unwanted signals entering the sidelobes.
4.7 Moving a Null to Reject Interference
An antenna pattern null has zero gain, so any signal arriving in the direction
of a null is not received. Adaptive, smart, or null steering antennas protect
RF systems from unwanted interference by moving nulls in the directions
of the interfering signals. These nulls reduce the received interference and
increase the signal to interference plus noise ratio (SINR) [28]. Factoring the
array factor polynomial as in (4.52) shows the null locations exist in the array
4.7 Moving a Null to Reject Interference
pattern. Moving one null to the direction of an interfering signal changes
one of the zeros in the array factor polynomial. Multiplying the factored
polynomial produces new wn that result in a new polynomial. Moving one or
more zeros in the array factor polynomial (nulls in the array factor) changes
all of the element weights (polynomial coefficients).
Example
A six-element uniform linear array with d = 𝜆/2 has zeros at
𝜓 = ±60∘ , ±120∘ , 180∘
n
The factored array factor is given by
AF = z5 + z4 + z3 + z2 + z + 1
∘
∘
∘
∘
∘
= (z − ej60 )(z − e−j60 )(z − ej120 )(z − e−j120 )(z − ej180 )
(4.55)
These zeros are shown on the unit circle in Figure 4.31. If a desired signal entering the main beam is 30 dB below a signal at 30∘ , then the interfering signal
dominates the desired signal, because the sidelobe level is only 13 dB below the
peak of the main beam (dashed line in Figure 4.32). Move a null to reject the
interference.
Solution
Moving a null in the array pattern to 30∘ mitigates the impact of the interfering
signal (see Figure 4.31). Follow these steps:
1. Picking the closest zero (𝜓 n = 120∘ ) and moving it to 𝜓 n = 180 ∘ sin(30∘ )
= 90∘ .
2. Put the new zero in (4.57).
3. Multiply the factored polynomial to get the new weights.
]
[
wn = 1∠0∘ 0.518∠ − 15∘ 1∠ − 30∘ 1∠0∘ 0.518∠ − 15∘ 1∠ − 30∘
Figure 4.31 The quiescent array factor for a
six-element array with half-wavelength
spacing. The null at 41.8∘ is moved to 30∘ in
order to cancel interference.
90
ψn = 90°
ψn = 120°
180
0
270
141
4 Antennas
10
Uniform
Null moved
Directivity (dB)
142
0
−10
−20
−30
−80
−60
−40
−20
0
20
40
60
80
θ (deg)
Figure 4.32 Zeros on the unit circle for a six-element array with half-wavelength spacing.
The null is moved from 𝜓 n = 120∘ to 𝜓 n = 90∘ in order to cancel interference.
Some of the element weights have both an amplitude and a nonzero phase.
Any of the other zeros could also have been chosen to move to 𝜓 n = 120∘ .
The resulting weights would be different. Old and new array factors appear in
Figure 4.32.
4.8 Null Filling
Sector antenna arrays on a tower or building increase antenna gain and
coverage on the ground by pointing the main beam at a slight angle toward
the ground. Tilting reduces interference between other antennas, since nearby
antennas do not point at each other. In addition, tilting creates a larger main
beam footprint on the ground that improves power density distribution to
users. Mechanically tilting the array or electrically steering the main beam
toward the ground (Figure 4.33). Mechanical tilt points a backlobe skyward.
Remote electrical tilt (RET) controls the mechanical tilt in real time in order
to change coverage. Phase scanning the main beam toward the ground steers
Mechanical tilt
Phase scan
Horizon
Nulls pointing at ground
Ground
Figure 4.33 Mechanical tilt and phase scan.
4.8 Null Filling
Tilt angle
Tilt angle
(a)
(b)
Figure 4.34 Tilted arrays: (a) Two 2-element and one 4-element linear array of folded
dipoles on Grouse Mt. in British Columbia, Canada and (b) Cell antennas on top of a Penn
State building in State College, PA.
the backlobe toward the ground (see right side of Figure 4.33). A high gain
backlobe might interfere with other antennas, so its pointing direction matters. The folded dipole arrays in Figure 4.34a tilt downhill. Figure 4.34b shows
examples of sector antennas on top of a building that tilt toward the ground.
Areas where nulls in the antenna pattern point at the ground (Figure 4.33)
create signal dead zones, so signal dropouts occur. These dropouts disrupt continuous, quality service. Null-filling eliminates the nulls in the antenna pattern
pointing toward the ground by moving array polynomial zeros off of the unit
circle. Zeros outside of the unit circle have an amplitude greater than one, while
zeros inside of the unit circle have an amplitude less than one.
Example
The eight-element phase scanned array in Figure 4.33 has three nulls pointing
at the ground. This array has seven zeros that lie on the unit circle (Figure 4.35)
[
]
𝜓 = ±45∘ ±90∘ ±135∘ 180∘
n
Employ null-filling by moving the appropriate zeros off the unit circle.
Solution
Follow these steps to fill in three nulls:
[
]
1. Moving three zeros 45∘ 90∘ 135∘ , off of the unit circle (Figure 4.35) by
increasing their amplitudes from 1.0 to 1.2:
∘
∘
∘
∘
∘
AF = (z − 1.2ej45 )(z − e−j45 )(z − 1.2ej90 )(z − e−j90 )(z − 1.2ej135 )
∘
∘
× (z − e−j135 )(z − ej180 )
143
144
4 Antennas
Figure 4.35 Three zeros moved off the unit circle
to fill nulls.
90
180
0
270
Figure 4.36 Null-filled pattern (solid) superimposed on
the uniform pattern (dashed).
Null fill
2. Multiply this factored polynomial to get the coefficients or weights:
[
]
w = 1 1 − j0.48 1.10 − j0.48 1.10 − j0.67 1.32 − j0.67 1.32 − j0.70 1.72 − j0.70 1.73
(4.56)
The array factors associated with both arrays appear in Figure 4.36. Filling the
nulls cost the array factor some of the main beam gain.
4.9 Multiple Beams
Some wireless systems need arrays with more than one beam in order to
communicate with multiple users or switch between beams. Four multiple
beam architectures appear in Figure 4.37 [22]. Figure 4.37a has an array with
M different corporate feed networks that produce M beams. Each beam
independently scans because each feed network has its own phase shifters. All
feed networks share the same elements and perhaps some other electronics
like filters and amplifiers. A software approach, called digital beamforming,
replaces much of the hardware in the corporate feed with ADCs on receive
(Figure 4.37b) and DACs (digital-to-analog converters) on transmit [29]. The
ADC converts the RF signal at each element into bits. Software then calculates
4.9 Multiple Beams
Elements
Elements
…
Amplifiers
…
…
Phase
shifters
1 : N divider
LNA
LNA
LNA
MXR
MXR
MXR
ADC
ADC
ADC
M
…
1 : N divider
2
1 : N divider
Computer
1
(a)
L2
L1
(b)
R1
Elements
R2
Hybrid couplers
45° phase
shifts
R1
L2
R2
(c)
L1
Beam 1 Beam 2 Beam 3
(d)
Figure 4.37 Different approaches to multiple beam arrays. (a) Multiple beam corporate
feed, (b) digital beamformer, (c) Butler matrix, and (d) Rotman lens.
the beams in the computer and independently steers them. A Butler matrix
generates 2M beams (M is an integer) using hybrid couplers and phase shifters
to create a hardware fast Fourier transform (FFT) (Figure 4.37c) [30]. The
beams point in fixed relative directions. A Rotman lens has M beams generated
from N elements (Figure 4.37d) by having M ports that transmit to/receive
from the N elements [31]. These beams also point in fixed directions. The
Butler matrix and Rotman lens are useful in beam switching strategies where
the beam with the best performance (e.g. SNR) is connected to the receiver or
transmitter while the others are not.
145
146
4 Antennas
4.10 Antennas for Wireless Applications
The application drives the type of antenna used in a wireless system. This
section describes some common wireless systems and their antennas.
4.10.1
Handset Antennas
A handset has three functional areas (Figure 4.38):
controller, I/O (input/output), and RF. Its brains
(the controller) contain the processor and memory.
The processor (microprocessors, microcontrollers,
and digital signal processing chips) monitors
data reception, selects receivers and transmitters,
adjusts impedance matching circuits, tunes filters,
and selects antennas. A hard disk drive, flash
memory, or random access memory (RAM) store
applications and data.
A user interfaces with a handset via an IO
(input–output) device, such as a touch screen, buttons, keyboard, microphone, speaker, vibrator, camera, sensor, LEDs (light-emitting diodes), biometric
sensors, and data ports.
The RF part of the handset generates, transmits,
and receives RF signals including:
•
•
•
•
•
Controller
Storage
Processor
IO
RF
GPS
Wi-Fi
Bluetooth
Cell phone
Near field comm
Antennas
Figure 4.38 Block diagram
of a handset with three
functional areas.
Global positioning system (GPS): 1575 MHz
Wireless local area network (WLAN) or Wi-Fi: 2.4 and 5 GHz
Bluetooth: 2.4 GHz
Voice and text: between 700 and 2700 MHz
Near field communications (NFC): 13.56 MHz.
Each frequency band has its own antenna. Sometimes a wideband or
multi-band antenna serves more than one frequency band. More frequency bands, including bands near 27 and 40 GHz, will be in use when 5G
(Appendix III) is adopted.
Initially, handsets had either a monopole or normal mode helical antenna,
or a combination of the two [32]. These antennas had main beams pointing
toward the user’s head which caused a high rate of RF energy absorption by the
head (Chapter 12). In 1996, the Hagenuk TCP6000 handset [33] introduced an
internal slot antenna for GSM (Global System for Mobile) handsets. Engineers
wanted the better performing exterior antenna, but consumers thought otherwise and eventually won. Table 4.3 compares the different types of handset
antennas. External monopole and helical antennas needed to be as short as possible or to preferably disappear. Moving antennas inside the handset requires
4.10 Antennas for Wireless Applications
Internal
External
Table 4.3 Comparison of handset antennas [32].
Antenna
Length
Advantages
Disadvantages
Monopole
𝜆/4
• Easy to manufacture
• Good performance
Helix
𝜆/8 − 𝜆/4
Monopole + helix
𝜆/4 − 𝜆/2
• Easy to manufacture
• Short
• Good bandwidth
See advantages of
monopole and helix
• Must extend and
retract
• Not multi-band
• Does not retract
• High SAR
Slot
𝜆/2
• Low SAR
• Big
• Not sensitive to hand • Single band
ceramic
𝜆/4
PIFA
𝜆/4
• Small
• Easy to surface
mount
• Low SAR
• Multi-band
PMA
𝜆/4 − 𝜆/2
• Very thin
• Similar to helix
See disadvantages of
monopole and helix
•
•
•
•
•
•
•
•
Heavy
Low gain
Narrow bandwidth
Low gain
Small bandwidth
Two contacts to PCB
High SAR
Large
them to coexist with all the other handset electronics inside the case. Handset
engineers developed clever designs made possible by modern manufacturing
technology. Figure 4.39 lists some of the technologies used to manufacture
handset antennas.
The first internal handset antennas were manufactured using stamped metal
or PCB processing [32]. Metal stamping molds a flat sheet of metal around
a die to get the desired shape. Hot stamping makes the antenna by placing
a metal foil in a heated die. This thin antenna then fits onto a molded part
in the case. Development and tooling costs are low. This approach works
when manufacturing large parts with simple geometry but not for thin lines
on 2D layouts or for 3D parts. More recently, molded interconnect devices
(MID) technology surpassed hot stamping. It integrates the antenna into the
handset mechanical housing [34]. This processing enables 3D layouts and
reduces component count and cost by integrating connectors, sockets, or
other devices. MID antennas, along with other circuits, are chemically or
mechanically etched from the thin copper layers that sandwich a dielectric
layer in a PCB. These inexpensive antennas have electronics and passive
devices co-integrated on the PCB. Unfortunately, this approach is limited to a
flat surface which limits the available area in the handset.
Improved manufacturing techniques, like two-shot molding, provide
cost-effective production of highly repeatable interconnect devices [35]. It
147
148
4 Antennas
Type
Stamped
metal
Theory
Stamped steel part
integrated together
with plastic piece
Advantages
Easy assembly,
versatile, spring clip
contacts can be
integrated into antenna
Copper has high
conductivity, can
contain air gap
(between antenna and
back cover) or mounted
on inside of cover to
maximum volume
Flex-film
Copper etched flexible
film glued onto plastic
piece
Hot Stamp
Uses heat and pressure
to place a metal part to a
surface (no 2-D curves
to bend over)
Stable pattern, no
pealing, similar
performance to copper
flexfilm
Antenna trace on PCB
In-expensive, no
assembly
PCB
trace
Disadvantages
Examples
Long lead times for
patterns, minimum line
width, cannot utilize
layered antennas or
3D curves
Requires several partsmore logistics, glue and
mechanical tolerances
Flat pattern only, no 3D
curves
Design limited to flat
surface, worse
performance due to
proximity of antenna to
PCB components
MID (Metal interconnect device)
One-shot:
3D
masking
or laser
etching
Two-shot
Uses a 3-D mask to print
the pattern onto a plastic
piece or laser to etch a
pattern in electroplated
metal
Two different kinds
of plastic are molded
together-metal adheres
to one plastic and not
the other
Shorter lead times for
antenna pattern
changes, stable patternno pealing
Expensive, antenna
pattern limited to outer
surface. not suitable for
mass production
Suitable for massproduction, stable
pattern-no pealing,
allows complete RF
design freedom
Expensive. PCB mount
only, long lead times
Figure 4.39 Comparison of handset antenna manufacturing techniques. Source:
Reproduced with permission of IEEE [32].
begins with a shape made from a plateable polymer, usually ABS (acrylonitrile,
butadiene, and styrene). Next, a second nonplateable material, usually polycarbonate, molds over top of the first shape leaving some areas of the first
material exposed. The two parts are then fused together before undergoing
electrolysis plating. In this step, the plateable plastic becomes metallized, while
the nonplateable plastic remains nonconductive. Two-shot has high initial
tooling costs, which limit its use to higher volume applications, because it has
limited design flexibility and long development times.
4.10 Antennas for Wireless Applications
Laser direct structuring (LDS) uses a laser to draw the antenna on the
surface of a plastic compounded with a special laser-sensitive metal complex
[35]. Exposing the polymer to the laser beam breaks down the metal complex
into elemental metal – either copper or palladium – and residual organic
compounds. The laser draws the antenna onto the part and leaves behind a
roughened surface containing embedded metal particles. These particles act
as nuclei for the crystal growth during subsequent plating with copper. Since
it uses a single material, LDS offers much lower cost and complexity than
two-shot molding. In addition, LDS draws extremely narrow traces compared
to hot stamping and two-shot molding. LDS offers more flexibility than other
MID processes because circuit redesigns only require reprogramming the laser
unit. The laser also performs the plastic surface preparation, unlike two-shot
molding, which requires an additional etching step to prepare the plastic for
plating.
Multi-band antennas inside the handset serve two or more frequency bands
in a smaller space than separate antennas for each band. The tri-band antenna in
Figure 4.40 covers NFC at 13.56 MHz as well as the cell bands between 700 and
2700 MHz [36]. Inductors L1 and L2 have high impedances at frequencies above
700 MHz and low impedances at frequencies below 100 MHz. Thus, the current
in loops 𝓁 2 and 𝓁 3 only flow at lower frequencies (NFC). Capacitor C 1 has a low
impedance at frequencies above 700 MHz and a high impedance at frequencies
below 700 MHz, so the loop 𝓁 1 only conducts at the higher frequencies in the
cell phone bands while L1 and L2 act as open circuits, so the antenna has the
characteristics of an inverted F antenna.
The Samsung Galaxy S4 antennas showcase MID technology in a handset
(Figure 4.41) [37]. All the antennas integrate into the plastic case, so that they
are compatible with other functional blocks and difficult to locate. Future
handsets will require an increasing number and range of frequencies and wider
bandwidths. Figure 4.42 shows the locations of the antennas in the handset of
the Samsung Galaxy S9 [38]. The NFC antennas are loops. One also serves as
a charging coil. The MST (magnetic secure transmission) antenna enables use
of Samsung Pay.
An interesting antenna problem arose when AT&T sold the Apple iPhone
4 in 2010 [39]. Apple’s new design had two antennas: (i) Wi-Fi, Bluetooth, and
Figure 4.40 Antenna
design that covers NFC as
well as the 700–2700 MHz
bands.
L1
l2
C1
l3
l1
Ground plane
L3
NFC
C2
700−2700 MHz
L2
149
4 Antennas
Wi-Fi and Bluetooth antenna
Figure 4.41 Antennas in a
Samsung Galaxy S4.
GPS
antenna
Contact for Wi-Fi and 4G antenna
Bluetooth antenna
3G/GSM antenna
GPS antenna
150
Figure 4.42 Antennas on a Samsung Galaxy S9.
MST antenna
NFC antenna/
wireless
charging coil
Main antenna
GPS: smaller antenna beginning in the bottom left and running to the top left
of the handset and (ii) voice and data antenna: much larger antenna running
around almost three quarters of the phone. Any conductor (like a hand) that
bridged the gap between the two antennas caused them to detune (not resonate
at the desired frequency). Naturally holding the bottom left corner of the
4.10 Antennas for Wireless Applications
Table 4.4 Signal attenuation (dB) for three different cell phones [39].
Cupping
tightly
Holding
naturally
On an
open palm
Holding naturally
inside case
iPhone 4
24.6
19.8
9.2
7.2
iPhone 3GS
14.3
1.9
0.2
3.2
HTC nexus one
17.7
10.7
6.7
7.7
Attenuation (dB)
handset makes skin contact between the two antennas resulting in significant
signal attenuation. Table 4.4 gives the signal attenuation associated with three
different cell phones around the time when the iPhone 4 debuted. Apple fixed
the problem by giving owners a nonconductive case that prevented a user’s
hand from directly touching the antennas. The case significantly improved
reception as shown by the last column in Table 4.4.
4.10.2
Cellular Base Station Antennas
A cellular wireless system divides a region into cells with a base station at
the center of each cell. The base station communicates with any mobile user
entering its cell. Ideally, the power radiated by an omnidirectional base station
antenna decreases equally in all directions around the antenna, in which
case the cell would be a disk shape. Assuming the cellular base stations are
omnidirectional in azimuth and there are no obstacles, a base station transmits
and receives from users in the areas designated by circles in Figure 4.43. Each
circle corresponds to the outer boundary beyond which the SNR drops too low
due to free space loss. The intersection of the circles of the six base stations
Figure 4.43 Coverage
area of base stations with
omnidirectional antennas.
Hexagon
Base
station
Base
station
coverage
area
151
152
4 Antennas
Ro
Cell (m,n)
Ri
Ro
Ro
n
Cell (0,0)
60°
Rf
m
Figure 4.44 Frequency reuse in a cellular system.
around the center base station form the apexes of a regular hexagon designated
by the dashed lines. As a result, a hexagon rather than a circle describes a
base station coverage area known as a cell (Figure 4.44). An inscribed circle of
radius Ri and a circumscribed circle of radius Ro bound a hexagon. The radii of
the inscribed and circumscribed circles are related by Ri = 0.866Ro . The cells
overlap in order that communication occurs seamlessly when a mobile unit
passes from one cell to another. A handoff occurs when a user moves from one
cell to another causing one base station to transfer the communication to a
new base station.
To minimize interference between cells, nearby cells operate in different frequency bands. A cellular system reuses a frequency band in a co-channel cell far
enough away to preclude interference. Frequency reuse depends on the size of
the cell and the number of available frequency bands. A cluster of cells contains
N cluster cells with no cell operating in the same frequency band. Clusters act like
puzzle pieces that assemble the coverage area of a cellular system. Increasing
N cluster reduces the number of clusters needed to cover a cellular service area
and reduces the system capacity [40].
Figure 4.44 divides the cellular region
into regular hexagons with a
√
center-to-center spacing of 2Ri = Ro 3. The cluster size (N cluster ) is the
number of cells in a fixed group in which each cell operates at a unique
frequency. This fixed group repeats throughout the cellular region in order to
allow frequency reuse but at the same time minimize interference between
cells. The ratio Rf /Ro determines the number of cells in a cluster.
{ ( )2 }
1 Rf
(4.57)
Ncluster = round
3 Ro
where Rf is the distance between two cells operating at the same frequency.
Table 4.5 has calculations of (4.57) for various ratios of Rf to Ro . Figure 4.45
4.10 Antennas for Wireless Applications
153
Table 4.5 Number of clusters calculated from the reuse distance and cell size.
Ncluster
3
4
7
9
12
13
16
19
21
25
27
Rf /Ro
3
3.5
4.6
5.2
6
6.2
6.9
7.5
7.9
8.7
9
1
1
8
10
4
2
7
2
3
1
2
8
3
1
3
4
4
(a)
18
15
6
11
3
13
3
10
2
9
2
17
14
5
12
6
9
1
11
5
4
13
4
19
16
6
12
(b)
(c)
Figure 4.45 Arrangement of cells for different values of Ncluster . (a) 4, (b), 13, and (c) 19.
shows possible cell arrangements when N cluster = 4, 13, and 19. A number in a
cell corresponds to a frequency band that differs from all other cell numbers.
Assume that (m, n) mark the cell centers along the m and n directions shown
in Figure 4.44, where m and n are integers. The co-channel distance from a cell
at (0,0) to cell (m, n) is given by [40]
√
Rf = Ro 3(m2 + mn + n2 )
(4.58)
Cell size depends on the geography and number of mobile users. Cell splitting creates smaller cells from a standard size cell. The layout in Figure 4.46
makes sense in flat terrain where cells extend several kilometers in radius.
Buildings, trees, and hills block signals and reduce the size cell as well as create
amoeba-like shapes rather than nice circles or hexagons. Cells in metropolitan
areas cover a significantly smaller area due to high traffic. In high-use areas,
designers divide cells into small cells with names like [41]
• Macrocell: Largest cell associated with a base station. Maximum range is
35 km.
• Microcell: Smaller cell in high population areas that are low-power. Maximum range is 2 km.
• Picocells: Provides very localized coverage (usually inside buildings). Maximum range is 200 m.
154
4 Antennas
• Femtocells: Smallest cell that provides coverage inside a room or a small
building. Maximum range is 10 m.
The lower right cell in Figure 4.46 shows how small cells fit within a larger cell.
A small cell antenna must have an appropriate location that covers the desired
area as well as a connection to the network.
Sector antennas are typically 1–2 m long arrays that have 10–20 dB of gain
[42]. The top right cell in Figure 4.46 shows a hexagon cell divided into three
120∘ sectors. Base station antennas have the best coverage when placed high
above ground on a tower (Figure 4.47) or a building (Figure 4.48). Figure 4.47 is
a tower that has several types of antennas. An omnidirectional antenna on top
has a 360∘ view of the area. Below the omni are three sets of sector antennas.
120°
Figure 4.46 Hexagonal cell sectors
which are further divided into micro
and pico cells.
120°
120°
Macro
Micro
Pico
Figure 4.47 Base station antenna tower.
Omni-directional
Sector
Microwave link
4.10 Antennas for Wireless Applications
Figure 4.48 Sector antennas on top
of a building.
Figure 4.49 A six-sector base station
antenna.
Several microwave dishes on the tower send and receive signals to/from distant locations. The building below the tower contains transmitters, receivers,
control units, cooling, and connections to cables that run underground. Large
towers can have several venders renting space for their antennas. Not all base
station antennas in the world divide the hexagon into three sectors. The base
station in Tasmania, Australia in Figure 4.49 divides the cell into six sectors.
Local ordinances sometimes require camouflaging antenna towers for
aesthetic reasons. Approaches to camouflage including putting base station
antennas in fake trees, cacti, and rocks as well as inside signs and attics.
Figure 4.50 has two examples of base station antennas mounted in fake trees.
155
156
4 Antennas
Sector antennas
(a) Annapolis, MD
Sector antennas
(b) Chengdu, China
Figure 4.50 Base stations camouflaged as a trees. (a) Annapolis, MDand (b) Chengdu, China
Companies pay business and home owners to install antennas on or inside
structures on their properties [43].
Handset users want to seamlessly move between outdoors and indoors with
no disruptions in call quality and service. An in-building cell includes amplifiers
to overcome path loss in the building and the internal cable losses. There are
two approaches to in-building systems [45]
• Extend an existing cell site by borrowing its signal and bringing it indoors.
This approach works when the existing cell site has sufficient capacity to handle the current and projected capacity.
• Locate a cell site or a portion of a cell site in the building. Subscriber traffic
must be high enough to justify on-site cellular PCS (personal communications service)/wireless/cellular equipment. The antenna needs a clear view of
the closest cell site and a coaxial cable that routes the signal to the in-building
system.
Either omnidirectional or leaky coaxial cable antennas distribute the signal
inside a building. Omnidirectional antennas work best in large unobstructed
areas, such as the interiors of shopping malls and convention centers. A leaky
coaxial cable is a better choice for restricted coverage, such as hospitals,
elevator shafts, corridors, subway tunnels, and office spaces.
4.10.3
Reflector Antennas
For most applications, the parabolic dish or reflector antenna serves as the
workhorse for long distance wireless communications. They convert a small
feed antenna into a much bigger aperture antenna. Figure 4.51 shows four
methods of illuminating a parabolic reflector surface. The front-fed reflector
4.10 Antennas for Wireless Applications
Figure 4.51 Four types of reflector
antennas.
(b) Cassegrain
Constant phase
(a) Front fed
(c) Gregorian
(d) Off-axis fed
places a feed antenna (typically a horn antenna) at the focal point of the
paraboloid. The distance from a horn antenna at the focal point to the surface
of the reflector then to a plane (constant phase) in front of the reflector is
the same for all angles (definition of a parabola). The horn feed blocks the
center of the reflector and has difficulty fully illuminating the surface of a
large reflector. Two other approaches use a horn that radiates from the center
of the main reflector to a small subreflector that then illuminates the main
reflector surface. Subreflectors enable feeds to provide a desired illumination
for a large reflecting surface. A Cassegrain reflector has a convex subreflector
(Figure 4.51b). A Gregorian parabolic reflector has a concave subreflector
(Figure 4.51c). The feed or subreflector along with its support structure blocks
front-fed, Cassegrain, and Gregorian main reflectors, so an offset design
removes the feed from the front of the aperture as shown in Figure 4.51c,d.
Front-fed reflectors usually use a Cassegrain design, because the subreflector
is closer to the main reflector than in a Gregorian design, making the antenna
more compact. Offset-fed reflectors often use Gregorian designs, because the
subreflector is closer to the main reflector in the vertical direction, and better
control of spillover is possible [46].
Extremely large Cassegrain or Gregorian reflectors use a beam-waveguide
[44] that has the feed horn and support equipment in a room far below the
antenna (Figure 4.52). The signal from the feed horn travels to the subreflector
via several reflecting mirrors. High-power, water-cooled transmitters and
low-noise cryogenic amplifiers do not have to tilt as the antenna moves. This
157
4 Antennas
Figure 4.52 Alfouvar satellite ground station
Cassegrain beam-waveguide reflector antenna
in Portugal.
Cassegrain reflector
Beam-waveguide
158
Equipment room
design provides easy access to system components. An example of a beam
waveguide Cassegrain reflector is the Alfouvar satellite ground station located
about 30 km North of Lisbon (Figure 4.52). It provides radio, telephone, and
television communications services to Portuguese speaking people around the
globe.
The offset feed eliminates feed blockage by using only a portion of the
parabolic reflector. In this case, the feed illuminates the reflector while being
out of the way of the reflected rays (off-axis feed in Figure 4.51d). This offset
reflector design is widely used in dish antennas for satellite television as shown
by the house mounted reflector in Figure 4.53.
Figure 4.54 has some examples of typical front-fed satellite communication
system reflector antennas. These antennas at the National Center for Atmospheric Research (NCAR) in Boulder, CO receive weather data from satellites.
Figure 4.53 Satellite TV offset
reflector antenna.
4.10 Antennas for Wireless Applications
(a)
(b)
(c)
Figure 4.54 Three types of satellite dishes at NCAR. (a) Wire mesh, (b) solid surface and (c)
solid surface with shroud.
The dish in Figure 4.54a has a wire mesh reflector surface. Operating at frequencies where the mesh spacing is less than 𝜆/10 minimizes the amount of
signal that passes through the mesh. The wire mesh reduces the weight as well
as the stress due to wind. The reflector in Figure 4.54b has a solid reflector. The
satellite dish in Figure 4.54c has a solid surface with a shroud around the edge
that suppresses sidelobes due to diffraction from the edges.
4.10.4
Antennas for Microwave Links
Terrestrial microwave antennas relay information from transmitter to receiver
over long distances. Microwave links have highly directional antennas for LOS
communications at microwave frequencies in order to overcome the large free
space loss. A microwave link is cheaper and faster to install than cables, because
it does not require laying cable over long distances. Figure 4.55 shows several
microwave link towers on top of Colorado Mines Peak (12 497 ft/3809 m).
These antennas have an unobstructed view for relaying microwave signals over
the Rocky Mountains.
Figure 4.55 Microwave link antennas on top of Mines Peak in Colorado.
159
160
4 Antennas
Reflector antennas work well in microwave links, because they have very
narrow beamwidths for high gain and resolution. The narrow beamwidths of
transmit and receive antennas require precise pointing, however. Small pointing deviations cause severe signal deterioration, so the antennas need mounts
on very stable, wind-resistant towers. If the frequency of a 244-cm diameter
antenna increases from 2 to 6.5 GHz, the maximum allowable tower motion
decreases from 3.5∘ to 1.0∘ which increases the tower rigidity requirement by
3.5 times [45]. A 2-ft diameter dish at 22 GHz has a 2∘ beamwidth and requires
even more stringent wind load specifications.
Another microwave link antenna, the Hogg or horn-reflector antenna, has
a portion of a parabolic antenna mounted at the mouth of a pyramidal horn
antenna (Figure 4.56) [47]. The reflector focus lies at the apex of the horn.
This design has low sidelobes, because nothing blocks the aperture and the
hood discourages radiation out of the sides. This very heavy, wind-resistant
horn-reflector requires a large support structure as shown in Figure 4.57a.
Its biggest use is for point-to-point communications where it does not have
to rotate. Since the 1970s the shrouded parabolic dish antennas have largely
replaced the Hogg antenna, because they have equally good sidelobe performance with a lighter more compact construction that can be mounted on
simple towers (Figure 4.57b). Flat or cone-shaped radomes over the dishes
increase the stability and protect the reflector surfaces (Figure 4.58).
θl =90−αo
x
X 2 + Y 2 = 4f (Z + f)
θl
ρ ξ
η
Projected
aperture
α
αo
El (Long polarization)
z
(a)
(b)
(c)
Figure 4.56 Horn reflector antenna. (a) Horn-reflector on Miners Mountain, CO (Figure 4.57)
(b) Diagram from patent [47]. (c) Conical horn reflector antenna. Source: Reprinted by
permission of Ref. [48]; © 2010 IEEE.
4.10 Antennas for Wireless Applications
Horn-reflector
(a)
Shrouded reflector
(b)
Figure 4.57 Horn-reflector antennas require hefty mounts compared with shrouded
reflector antennas (Miners Mt, CO).
Figure 4.58 Radomes that protect microwave link antennas. The two antennas on the let
have conical radomes while the one on the right has a flat radomes over a shroud.
Example
Estimate the loss in dB of a 1 m dish transmitting antenna operating at 5 GHz
that is 1 km away from the receive antenna. The transmit antenna has a pointing
error of 1∘ while the receive antenna has no pointing error.
Solution
Start with (4.21) when 𝜆 = 3 × 108 /5 × 109 = 0.06 m and z0 = 1000 m
161
162
4 Antennas
r0 = 1000 tan 1∘ ≈ 17.5 m for a 1∘ pointing error using the small angle approximation for tan
(
)
kr r
J1 za 0
0
AP ≃
kr r
a 0
z0
kra r0
=
z0
2𝜋
(1∘ ) 𝜋 ∘ (17.5)
0.06
180
= 0.032
1000
{
}
}
{
J1 (0.032)
0.016
= −20 log{0.5} = 6 dB
Loss ≃ −20 log
= −20 log
0.032
0.032
This high loss due to a small pointing error emphasizes the importance of aligning transmit and receive antennas as well as mounting the antennas on solid
support structures.
4.11 Diversity
Diversity in the base station mitigates the impact of signal amplitude drops
known as signal fading. If a transmitted signal has multiple, independent paths,
then the receiver lies in a region of high amplitude variations. The receive
antenna either selects the largest signal or combines the signals in a way
that enhances reception. Most diversity schemes require multiple antennas.
Each additional antenna contributes to the real estate problem on handsets
but not as much on base stations. Diversity gain is the increase in SNR due
to a diversity scheme. Diversity means receiving a signal at multiple places,
frequencies, polarizations, or times and picking the best one. It works best
when the signals at the different places, frequencies, polarizations, or times
are not correlated.
4.11.1
Spatial Diversity
Spatial diversity requires two or more receive antennas separated in space.
Figure 4.59 shows the received power density distributed over one dimension
in space. Some regions have low signal power due to fading. An antenna in
a region of fading does not receive enough signal for detection. The three
antennas in Figure 4.59a all receive low signal power. The antennas are close
together, so they lack enough spatial diversity (separation) in order to have
one of the antennas in a region of high signal power. Spatial diversity increases
by adding more antennas or by putting more space between the existing
antennas. Figure 4.59b demonstrates that increasing the antenna separation
distance puts at least one of the antennas in a high signal power area. The
Received power (dBm)
Received power (dBm)
4.11 Diversity
Position (m)
Position (m)
(a)
(b)
Figure 4.59 Spatial diversity increases the odds of an antenna receiving a strong signal.
Figure 4.60 Wireless routers with multiple monopoles for spatial diversity.
wireless routers in Figure 4.60 have more than one monopole in order to
increase spatial diversity. When multipath signals come from all directions,
antenna spacing between 0.5𝜆 and 0.8𝜆 results in independent (decorrelated)
channels. Spatial diversity requires a significant antenna separation in the VHF
and lower UHF bands, because the effectiveness of the antenna separation
depends on 𝜆.
Selection diversity picks the strongest signal out of N received signals in an
antenna array. For instance, the left most antenna in Figure 4.59b receives the
highest signal power, so it is selected as the active antenna while the other
two are ignored. The average selection diversity gain for N independent and
Rayleigh distributed signals is [49]
Gsd =
N
∑
1
n
n=1
≈ 𝛾E + ln N + 0.5∕N
(4.59)
where 𝛾 E = 0.5772 K is Euler’s constant. Each additional element contributes a
smaller amount to Gsd . The average increase to the SNR is given by
SNR = (𝛾E + ln N + 0.5∕N)SNRn
(4.60)
163
164
4 Antennas
where SNRn is the average SNR at one antenna element. Selection diversity only
requires measuring the SNR at each element but not the amplitude and phase
of the signals at the elements (Figure 4.61).
Rather than just selecting the antenna with the highest signal power, maximum ratio combining (MRC) adds the received signals together after equalizing
the element weights in the array. It has the optimum statistical fading reduction
compared to all other linear diversity combiners. The diversity combiner SNR
equals the sum of the SNRs of the signals from each antenna. On average, the
diversity gain of MRC is
(4.61)
GMRC = N
which beats Gsd . The average SNR of MRC is
SNR = NSNRn
(4.62)
MRC requires measuring the amplitude and phase of the signals at the
elements. MRC has trouble keeping up with weight changes in fast varying
channels.
Equal gain combining (EGC) equalizes the phases but keeps the amplitude
weights uniform to improve the SNR. On average, the EGC diversity gain is
GEGC = 1 + (N − 1)
𝜋
4
(4.63)
which is very close to GMRC and significantly better than Gsd . The average SNR
of MRC is
]
[
𝜋
SNR = 1 + (N − 1) SNRn
4
(4.64)
EGC requires a measurement of the signal phase at each element.
Channel Antennas Receiver
Path 1
Transmitter
Path 2
Signal
Path N
Diversity
combiner
or
switching
network
Signal
Figure 4.61 Diversity exploits the
different paths a transmitted signal
takes to arrive at the receiver.
4.11 Diversity
4.11.2
Frequency Diversity
f1
a3
f2
Frequency diversity switches between multiple frequency bands to until it finds the band that increases
a2
the total signal power. For the same transmitting and
receiving antenna locations, the peaks and nulls due f3
to multipath occur at different locations for different
frequency channels. Figure 4.62 shows the signal level
a1
at a receive antenna for three different frequencies.
Switching from f 1 to f 2 or f 3 significantly improves the
received signal level from a1 to a2 to a3 . The frequency Figure 4.62 Frequency
separation that yields independent fading on the differ- diversity means
picking the
ent channels depends on the path lengths. Large path frequency which
length differences require small frequency differences results in the
in the channels. Microwave line of sight links use highest signal level.
frequency diversity. A frequency diversity system that
experiences an SNR decrease at one frequency switches to a backup frequency
band that has a better SNR.
4.11.3
Polarization Diversity
Since reflections change the polarization of a signal, a transmit antenna
with one polarization delivers two orthogonal polarizations to the receiver.
Polarization diversity has two antennas with orthogonal polarizations. The
receiver selects the antenna with the strongest signal. Cellular base stations
use polarization diversity. Figure 4.63 shows two approaches to polarization
diversity. The first has an array of vertically polarized antennas next to an
Figure 4.63 Polarization diversity has
adjacent antenna arrays with
orthogonal polarization.
165
4 Antennas
z-polarization
Figure 4.64 Polarization diversity
antennas on a computer.
y-polari
lar
iza
tio
n
zation
xpo
166
array of horizontally polarized antennas. The more common configuration
on the right uses slant 45∘ polarization diversity, because both polarizations
contain horizontal and vertical components (Fresnel reflection coefficients are
polarization dependent). One slant polarization duplexes transmit and receive,
while the other slant polarization only receives. In this way, one polarization
diversity antenna replaces two vertically polarized antennas spaced several
feet apart. Decreasing the number and size of the antennas decreases the
tower loading as well as the cost of renting space on the tower. Two orthogonal
polarizations mean that polarization diversity has only two channels that lack
the independence found in spatial and frequency diversity but are independent
enough to enhance performance. Polarization diversity is simple and cheap to
implement. Figure 4.64 is an example of polarization diversity on a computer
wireless card using three orthogonal monopoles.
4.11.4
Time Diversity
Time diversity transmits the same signal at multiple time slots separated by
at least the coherence time of the channel [49]. Coherence time is the period
in which the channel impulse response does not change. Outside the coherence time, multiple versions of the same signal encounter independent fading
conditions on their way to the receiver, because channel conditions change.
Problems
4.1
An antenna has a pattern given by AP = sinc(10 sin 𝜃) when
−90∘ ≤ 𝜃 ≤ 90∘ for (a) AP rectangular plot, (b) |AP| rectangular
plot, (c) 20log|AP| rectangular plot, and (d) |AP| polar plot.
Problems
4.2
Calculate the directivity of an antenna when AP = cos 𝜃 for 0 ≤ 𝜃 ≤ 𝜋/2
and zero elsewhere.
4.3
Find the directivity of an antenna with AP(𝜃) = sin 𝜃.
4.4
An antenna has 𝜃 AP(𝜃) = sin(5𝜋 sin 𝜃)/(𝜋 sin 𝜃) for 0 ≤ 𝜃 ≤ 𝜋/2 and zero
elsewhere. Find the gain if the aperture is 95% efficient.
4.5
Estimate the gain of a 50-m diameter antenna at 1 GHz.
4.6
An antenna operates over all frequencies in X band, what is its bandwidth?
4.7
Find the percent power received by a linearly polarized antenna when a
LHP circular signal is incident.
4.8
Plot the PLF vs. rotation angle when a linearly polarized wave is incident
on a linearly polarized antenna that rotates in a plane perpendicular to
the incident field.
4.9
The IEEE defines the far field of an antenna as R = 2D2max ∕𝜆 for
ΔR ≤ 𝜆/16. Low sidelobe antennas require greater phase accuracy. What
should the far field distance be if ΔR ≤ 𝜆/48?
4.10
Find the equation for the electric field of a 𝜆/2 long dipole and plot the
antenna pattern in dB.
4.11
Use MATLAB to calculate and plot the 3D antenna pattern of a 𝜆/2
dipole.
4.12
Find the radiation resistance of a loop with (a) 1 turn and (b) 100 turns
when r𝓁 = 𝜆/25.
4.13
Find the magnitude of the relative electric field of a small loop
(r𝓁 = 0.01𝜆) as a function of 𝜃.
4.14
Plot AR of an axial mode helix in dB vs. N (from 1 to 20).
4.15
Assume a 12 cm × 6 cm with a uniform field projects onto a flat screen
that is 20 cm away at 10 GHz. Plot the relative antenna pattern over a 2 m
by 2 m area that is centered 20 cm from the aperture.
167
168
4 Antennas
4.16
A rectangular patch on a substrate with 𝜀r = 2.2 that operates at 2.5 GHz.
Plot: (a) E𝜃 vs. 𝜃 𝜙 = 0∘ and 90∘ , (b) E𝜙 vs. 𝜃 𝜙 = 0∘ and 90∘ .
4.17
Plot L and W for 1 ≤ 𝜀r ≤ 10 when the patch resonates at 2.4 GHz.
4.18
Consider an eight-element uniform array along the x-axis with an element spacing of 1.5 cm at 10 GHz. What is the phase shift needed at each
element to steer the main beam to 30∘ .
4.19
Plot the array factors on one graph in rectangular format for an
eight-element uniform array when the element spacing is 0.5, 1.0, and
2.0 wavelengths.
4.20
On one graph, plot the binomial array factors when N = 2 through nine
elements.
4.21
Derive the Chebyshev weights for a six-element array with peak sidelobes that are 20 dB below the main beam. Plot the array factor.
4.22
Derive the Taylor weights for a 20-element array with peak sidelobes that
are 20 dB below the main beam and n = 5. Plot the amplitude weights
and array factor.
4.23
Calculate the array weights shown in Figure 4.29 and the corresponding
array factors in Figure 4.30.
4.24
Plot the array factor and unit circle representation of a four-element uniform array when the element spacing is 0.5 wavelengths. Now, move the
two zeros at ±90∘ to ±120∘ . Plot the unit circle representations and array
factors before and after moving the nulls. What are the new amplitude
weights? Plot the array factors in dB on a rectangular plot.
4.25
Start with a Taylor n = 4 sll = 25 dB taper and place a null at u = 0.25
when d = 0.5𝜆. Do not allow complex weights. Plot the unit circle representation, the array weights, and the nulled array factor superimposed
on the quiescent array factor.
4.26
A four-element array has roots at z = − 1 and ± j. Move the zeros to (a)
z = − 1.2 and ± 1.2j and (b) z = − 1.4 and ± 1.4j.
4.27
Calculate the weights for the array in the previous problem when the
zeros are moved to z = − 0.8 and ± 0.8j.
Problems
4.28
Use null-filling for linear array along the x-axis with six elements spaced
half a wavelength apart for 𝜃 between 0∘ and 90∘ . Try to totally eliminate
those nulls. Is it possible? Plot the unit circle representation and the array
factor.
4.29
Use null filling on an eight-element array with half wavelength spacing
along the x-axis for 𝜃 between 0∘ and 90∘ . Compare and contrast moving
the zeros inside and outside the unit circle.
4.30
An eight-element uniform array along the x-axis has interference incident at 𝜃 = − 38∘ . Move the array factor null at 𝜃 = − 48.6∘ so that it
points at the interference.
(a) Calculate the new array weights.
(b) Plot the unit circle representation.
(c) Plot the adapted array factor.
4.31
Repeat the previous problem but also move the null at 𝜃 = 48.6∘ to
𝜃 = 38∘ .
(a) Calculate the new array weights.
(b) Plot the unit circle representation.
(c) Plot the adapted array factor.
4.32
Derive (4.58).
4.33
Derive N cluster = m2 + mn + n2 from (4.57).
4.34
A reflector antenna has an f/D (focal length to diameter ratio) of 1.0. A
dipole antenna is placed at the feed facing the bottom of the paraboloid
reflector. Calculate the field strength at the bottom and edge of the reflector in the plane containing the focal point and the paraboloid vertex
when the dipole lies perpendicular to that plane.
4.35
A reflector antenna has an f/D (focal length to diameter ratio) of 1.0.
A small dipole antenna is placed at the feed facing the bottom of the
paraboloid reflector. Calculate the field strength at the bottom and edge
of the reflector in the plane containing the focal point and the paraboloid
vertex when the dipole lies in that plane.
4.36
Two mountain tops are 10 km apart operate at 7 GHz. The transmitter on
one mountain is a Hogg antenna that has an aperture of 2 m by 2 m. The
receive antenna on the other mountain is a 2 m dish antenna. Estimate
the pointing error in degrees that results in a 3 dB loss in power.
169
170
4 Antennas
4.37
On the same figure, plot the selection diversity gain vs. N using the exact
and approximate formula in (4.59).
4.38
On the same figure, plot the average SNR for MRC vs. N for SNRn = 5,
10, 15, and 20 dB.
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173
5
Propagation in the Channel
A water channel directs water from a source to a destination. Smooth channels
produce easily predictable laminar flow. Obstacles like rocks induce turbulence
that makes flow predictions in the channel difficult. A similar situation exists for
the flow or channeling of a radio frequency (RF) signal from a transmitter to a
receiver. In free space, the signal propagates from the transmitter to the receiver
with a loss proportional to the square of the distance traveled. Unfortunately,
the RF channel usually has many obstacles like buildings, cars, trees, etc. that
attenuate, reflect, refract, and diffract the signal along several different paths. In
addition, the transmitter and/or receiver motion induces Doppler shift in the
signal. The simple idea of free space loss needs modification or augmentation
to get accurate results. Interference and noise corrupt the signal and cause the
receiver additional problems.
Geometrical optics (GO), or ray tracing, typically models signal propagation
in a channel. Figure 5.1 illustrates some of the mechanisms encountered by
a transmitted signal (ray) traveling through a channel to the receiver. In general, a satellite system has a pure line of sight (LOS) channel. The LOS signal
travels a long distance through the atmosphere which attenuates, refracts, and
depolarizes it. These effects highly depend on the signal frequency. Diffraction,
reflection, and shadowing significantly impact signals in the channel. Statistical
models approximate complex relationships in a practical setting with relatively
simple stochastic models.
This chapter introduces RF propagation modeling in a channel. Approaches
to modeling a channel include measured impulse responses, electromagnetic
models, and statistical models. Determining the channel impulse response
requires experimental measurement of electromagnetic waves propagating in
the channel. Channel measurements require (i) expensive equipment, (ii) a
significant time investment, and (iii) complex setup. Realistic electromagnetic
computer models based on Maxwell’s equations help with siting antennas
and analyzing specific scenarios. Numerical approximations, like ray tracing,
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
174
5 Propagation in the Channel
Figure 5.1 Ray tracing in the signal propagation channels.
accommodate large, complex channels. Accurate electromagnetics models
are expensive and time-consuming. Several statistical models developed
from measurements provide simple estimates in place of more accurate
measurements or electromagnetic models.
5.1 Free Space Propagation
Free space propagation assumes a signal propagates in a vacuum with no
obstacles. Even the path between a satellite and a ground station has obstacles,
including reflections and diffraction from the satellite and ground. The LOS
communications link in Figure 5.2 has minimal obstacles. The transmitter
generates Pt which the antenna magnifies with gain Gt . Assuming that the
transmitted signal travels in all directions from the antenna means that the
power density at the receive antenna is Pt Gt /4𝜋R2 . If the receiver has an
effective aperture Ae , then the received power is
Pr =
Pt Gt Ae
4𝜋R2
(5.1)
The Friis transmission formula (5.1) serves as the fundamental model for a
communication link [1].
5.2 Reflection and Refraction
Ae
PtGt
4πR2
R
PtGt
Pr =
Pt
PtGt Ae
4πR2
Figure 5.2 Friis transmission formula.
A more realistic version of (5.1) includes losses lumped into a constant (L)
and a power loss due to distance (𝛾).
Pr =
Pt Gt Ae L Pt Gt Gr c2 L
=
4𝜋R𝛾
(4𝜋f )2 R𝛾
(5.2)
where Ae = G𝜆2 /(4𝜋). This chapter explains how to find appropriate values
for L and 𝛾 that depend on the propagation channel characteristics. The Friis
equation forms the foundation of the link budget of a wireless system. Typically,
the link budget analysis converts (5.2) into dB:
10 log(Pr ) = 10 log(Pt ) + 10 log Gt + 10 log Gr + 20 log 𝜆 + 10 log(L)
− 10𝛾 log R − 20 log(4𝜋)
(5.3)
The received power must exceed the receiver sensitivity in order for meaningful communications to occur.
5.2 Reflection and Refraction
A signal encounters obstacles as it travels through the channel. These obstacles
reflect and refract the signal. By the time the signal reaches the receiver, its
amplitude, phase, and duration differ from the transmitted signal. Updating
the Friis formula begins with calculating the effects of these reflections and
refractions on the signal.
An electromagnetic signal incident on a large perfectly flat medium (Ei ,
Hi , 𝜃 i ) reflects from (Er , Hr , 𝜃 r ) and transmits into (Et , Ht , 𝜃 t ) the medium as
shown in Figure 5.3. The plane of incidence contains the incident wave vector
and the normal to the boundary (dashed line in Figure 5.3). Any plane wave
175
176
5 Propagation in the Channel
TE polarization
TM polarization
Ei
Ei
Hi
θi
θr
Ht
Er
Hr
Et
ni
Hi
Hr
θi
nt
θt
θr
Er
θt
Ht
Et
Figure 5.3 Two orthogonal polarizations incident on the boundary between two media.
The plane of incidence contains the incident vector and the dashed line that is normal to the
surface.
consists of two orthogonal polarizations: transverse magnetic (TM) has the
electric field parallel to the plane of incidence and transverse electric (TE) has
the electric field perpendicular to the plane of incidence. Boundary conditions
determine the directions as well as the amplitude and phase of the reflected
and transmitted electric and magnetic fields.
The incident, reflected, and transmitted angles obey Snell’s law of reflection
𝜃i = 𝜃r
(5.4)
and Snell’s law of refraction
ni sin 𝜃i = nt sin 𝜃t
(5.5)
where
√
n = 𝜀r 𝜇r = index of refraction
𝜀r
= relative permittivity
𝜇r
= relative permeability
and the subscripts i, r, and t refer to the incident, reflected, and transmitted
waves. Reflection and transmission depend on the polarization of the signal as
well as the material properties and angle of incidence. The Fresnel reflection
coefficients for the two polarizations are different:
E0r
n cos(𝜃i ) − ni cos(𝜃t )
= t
E0i
nt cos(𝜃i ) + ni cos(𝜃t )
E0r
ni cos(𝜃i ) − nt cos(𝜃t )
=
=
E0i
ni cos(𝜃i ) + nt cos(𝜃t )
ΓTM =
(5.6)
ΓTE
(5.7)
The Fresnel transmission coefficients likewise depend on polarization.
TTM =
E0t
2nt cos(𝜃i )
= 1 − ΓTM =
E0i
ni cos(𝜃t ) + nt cos(𝜃i )
(5.8)
5.2 Reflection and Refraction
TTE =
E0t
2ni cos(𝜃i )
= 1 − ΓTE =
E0i
ni cos(𝜃i ) + nt cos(𝜃t )
(5.9)
These coefficients play an essential role in calculating the amplitude and phase
of the signal at the receiver.
Under certain conditions, a plane wave totally reflects from or totally transmits into a flat interface between two media. When ni > nt , The critical angle,
𝜃 c , occurs when the signal totally reflects from the surface.
( )
nt
−1
(5.10)
𝜃c = sin
ni
The entire signal incident at 𝜃 ≥ 𝜃 c reflects from the surface, while signals
incident at 𝜃 < 𝜃 c have partial transmission and reflection. The critical angle is
polarization independent.
No reflection (total transmission) occurs at the Brewster angle when a TM
polarized wave impinges on a boundary between two media
( )
nt
−1
𝜃b = tan
(5.11)
ni
Unlike 𝜃 c , 𝜃 b exists only for TM polarization. In addition, total transmission
only occurs at exactly 𝜃 b . Angles above and below 𝜃 b have some reflection. A
Brewster angle exists on both sides of the medium. In fact, the two Brewster’s
angles on either side of the interface sum to 90∘ .
( )
( )
nt
ni
90∘ = tan−1
+ tan−1
(5.12)
ni
nt
Table 5.1 summarizes the properties of the critical and Brewster angles.
Oftentimes, a signal passes through several layers of media as shown in
Figure 5.4. Buildings, windows, and the atmosphere have multiple layers with
different electrical properties. Calculating the reflection and transmission
coefficients for each layer requires an iterative process [2]. A single dielectric
slab of height h in free space has a relatively simple expression for the reflection
coefficient at normal incidence (𝜃 i = 0∘ ).
Γ=
Γd − Γd e−j2kd h
where Γd =
(5.13)
1 − Γ2d e−j2kd h
√
1− 𝜀r
√
1+ 𝜀r
and kd =
2𝜋
√ .
𝜆 𝜀r
Example
What is the minimum height of a dielectric slab in which the reflection at normal incidence is zero?
177
178
5 Propagation in the Channel
Table 5.1 Critical and Brewster angles.
Critical angle
Brewster angle
Characteristic
Total reflection
when 𝜃 ≥ 𝜃 c
Total
transmission
when 𝜃 = 𝜃 b
Polarization
TE and TM
TM
Index of refraction
ni > n t
θi = θc
Diagram
ni ≥ nt or ni ≤ nt
Ei
θr = θc
Hi
θi = θb
θt
Et
Ht
θi
θr
θt1
θi1 θr1
θt2
Layer 1
θi2 θr2
Layer 2
θi3
Layer 3
θt3
θt
Figure 5.4 Ray tracing through multiple layers.
Solution
No reflection from the slab means that Γslab = 0⇒ Γd − Γd e−j2k1 h = 0 so e−j2k1 h =
n
1 which only occurs when 2k 1 h = 2n1 𝜋 or h = 21 𝜆d .
5.3 Multipath
In phase
+
=
180° out of phase
+
=
+
=
135° out of phase
Figure 5.5 Adding one signal to another of the same frequency but different phases.
5.3 Multipath
Reflection changes the path of a signal as noted by the Fresnel equations. When
the reflected signal interacts with the LOS signal, interference patterns form.
In-phase signals add in amplitude as shown in Figure 5.5, otherwise the resulting sum is less than the sum of the two amplitudes. In fact, when they are out
of phase by 180∘ , they cancel or fade.
Multipath means that a transmitted signal arrives at the receiver by more
than one route. Figure 5.6 depicts a communication channel with a direct or
LOS signal plus one reflected or multipath signal. Since the signals take different paths to the receiver, they have different amplitudes due to reflections and
diffractions as well as additional free space loss. The phase and time of arrival of
each signal at the receiver differ, because their paths have different lengths. All
the signals sum to create a dispersed signal (longer duration than transmitted
τ
Transmit
antenna
RM
Re
fl
ec
ted
Direct
RLOS
τ
Receive
antenna
τ + Δτ
Figure 5.6 Multipath occurs when the signal takes more than one route to get to the
receiver. The received signal shows dispersion or broadening of the pulse.
179
5 Propagation in the Channel
signal) resulting from the time delays of the various paths. If the signal takes
two paths of length RLOS and RM (RM > RLOS ), then the multipath signal arrives
Δ𝜏 = (RM − RLOS )/c after the LOS signal.
Example
Use MATLAB to demonstrate dispersion with an 8.33 ns pulse at a carrier frequency of 2.4 GHz. There are three multipath signals that travel an extra 0.27,
0.97, and 1.29 m. The amplitude of these signals at the receiver is 0.7, 0.5, and
0.2 V compared to 1 V of the LOS signal. Plot the sum of these signals.
Solution
The multipath signals are delayed by 0.90, 3.23, and 4.30 ns. Adding the signals results in the plot in Figure 5.7 which has the received dispersed pulse
(solid line) superimposed on the LOS signal (dashed line). The dispersed pulse
is 4.3 ns long or 52% longer than the LOS pulse.
When two signals from different directions collide, they add together to form
an interference pattern like the one in Figure 5.8. A fade occurs when the signal amplitude drops below the receiver sensitivity level and is not detected. If
the sources move, then the interference pattern changes. A receiver moving
through the interference region experiences fluctuations in the received signal
amplitude.
Signal start times
0.0 0.9
3.2
Signal end times
4.3
8.3 9.2
1.5
11.5 12.6
LOS + multipath
LOS signal
1
Amplitude (V)
180
0.5
0
–0.5
–1
–1.5
0
2
4
6
t (ns)
8
10
12
Figure 5.7 Received signal for LOS only (dashed) and LOS plus multipath (solid).
5.4 Antennas over the Earth
Signal 1
+
Signal 2
Signal 1
Signal 2
Figure 5.8 Two electromagnetic signals at the same frequency but from different directions
meet and add together to produce interference with high- and low-signal amplitudes.
5.4 Antennas over the Earth
Figure 5.9 presents a microwave link between transmit and receive antennas
placed on towers. The LOS path (RLOS ) starts at the transmitter mounted ht
above the ground and ends at the receiver mounted hr above the ground. The
transmit antenna appears to have an image antenna beneath the ground that
radiates the reflected signal. Since the ground is not a perfect reflector, the
image antenna radiates a reduced amplitude compared to the actual antenna.
Using the Friis transmission formula, the received signal, sr (t), equals the sum
of the electric fields of the LOS and image (reflected) signals.
|
|
|
|
|
|
|
|√
−jkR
−jkR
√
|
LOS
i |
e
e
|
|
|ET | ∝ | Gt (𝜃t1 )Gr (𝜃r1 )
+ Γg Gt (𝜃t2 )Gr (𝜃r2 )
4𝜋RLOS
4𝜋Ri ||
|
|⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟|
|
|
|
|
LOS
Image
|
|
−jkR
√
|
LOS |
e
|F
≈ || Gt (𝜃t1 )Gr (𝜃r1 )
4𝜋RLOS || pg
|
where
Gt , Gr
RLOS
= gains
√ of the transmit and receive antennas
=
d2 + (ht − hr )2
(5.14)
181
182
5 Propagation in the Channel
Receive
θr2
θr1
Transmit
RLOS
θt1
hr
Ri
θt2
ht
ψg
d
Image
Figure 5.9 Diagram of direct and reflected rays.
√
Ri
=
d2 + (ht + hr )2
Γg
= ground reflection coefficient
RLOS
≈ Ri in the amplitude terms
Fpg = 1 + Γg e−jk(Ri −RLOS ) = path gain factor
d
= separation between transmitter and receiver.
The path difference between the LOS and reflected signals is
Ri − RLOS ≈
2ht hr
d
(5.15)
which delays the reflected signal by
𝜏=
2hr ht
cd
(5.16)
As d gets large, 𝜓 g (angle between the Earth and incident and reflected rays)
gets small, so from (5.6) and (5.7), Γg → ± 1. The path gain factor describes the
relative signal variation as a function of tower separation and tower heights [3]:
Fpg
)
(
⎧2 ||sin khr ht || = 2 ||sin(kh tan 𝜓 )|| TE polarization
t
g |
|
d
|
⎪ |
|
≈ |1 ± e−jk2h1 h2 ∕d | = ⎨ ||
)|
(
⎪2 ||cos khr ht || = 2 ||cos(kht tan 𝜓g )|| TM polarization
d
⎩ |
|
|
|
(5.17)
5.4 Antennas over the Earth
when 𝜓 g = 𝜃 t2 = 𝜃 r2 . The received power is a function of the tower heights and
their separation distance.
Pr = Pt Gt Gr
h2t h2r
d4
(5.18)
√
When d ≫ ht hr , then the power decreases as 1/d4 which far exceeds free
space loss. Note that the received power and path loss become independent of
frequency at large values of d. The path loss in dB is given by
LdB = 40 log d − 10 log Gt − 10 log Gr − 20 log ht − 20 log hr
(5.19)
Example
Plot the loss for a 2.4 GHz signal when the transmitter and receiver separation
distance increases from 100 m to 100 km for
(a)
(b)
(c)
(d)
free space
ht = 30 m, hr = 2 m
ht = 30 m, hr = 10 m
ht = 30 m, hr = 30 m
Assume the antennas are isotropic point sources.
Solution
Figure 5.10 shows the resulting plots. Up to a distance called the breakpoint,
the loss approximately equals that of free space (1/d2 ). After the breakpoint the
decay increases to 1/d4 . This behavior is typical in a multipath environment. In
fact, more than one multipath signal produces a 1/d𝛾 loss where 1.5 ≤ 𝛾 ≤ 4 and
𝛾 is the path loss exponent.
1
d2
Loss (dB)
–60
1
d4
–80
hr = 2 m
hr = 10 m
hr = 30 m
Free space
–100
–120
–140
20
25
30
35
10 log10 (d)
40
45
50
Figure 5.10 Propagation loss due to one bounce between two towers as a function of
tower height (hr ) and distance between the towers (d) when ht = 30 m.
183
184
5 Propagation in the Channel
The reflection coefficient in (5.14) depends on the type of ground (desert,
pavement, farm, etc.) and is given by [3]
√
⎧
⎪ −𝜀r (f ) sin 𝜓g + 𝜀r (f ) − cos2 𝜓g
⎪
√
⎪ 𝜀 (f ) sin 𝜓 + 𝜀 (f ) − cos2 𝜓
g
r
g
⎪ r
Γg = ⎨
√
⎪ sin 𝜓g − 𝜀r (f ) − cos2 𝜓g
⎪
√
⎪
sin
𝜓
+
𝜀r (f ) − cos2 𝜓g
⎪
g
⎩
TE polarization
(5.20)
TM polarization
where 𝜀r (f ) = 𝜀′r (f ) − j𝜀′′r (f ) = relative permittivity of the ground at frequency
f . Values of 𝜀r for various surface media as a function of frequency appear in
Figure 5.11 [4]. The plots reveal that the soil water content dramatically changes
both the real and imaginary parts of the relative dielectric constant. Also, fresh
water and salt water have much higher dielectric constants than snow. This
observation makes sense, because snow contains more air than water, so the
permittivity approximately equals the weighted average of the permittivities of
water and air (𝜀r ≈ 1).
The reflection coefficient for TM polarization is less than or equal to the
reflection coefficient for TE polarization at any incident angle. As a result,
when the incident wave has electric field components that are parallel and
perpendicular to the plane of incidence, the polarization of the reflected
wave differs from the polarization of the incident wave. For example a circularly polarized incident wave, becomes elliptically polarized after reflection,
because the orthogonal components of the circular polarization have different
reflection coefficients.
Example
A 2 GHz right-hand circular polarization (RHCP) wave is incident on flat
ground in which the soil has 0.1 moisture content at 𝜓 g = 30∘ . What is the
axial ratio of the reflected wave?
Solution
′
′′
From Figure 5.11, the 𝜀√
r ≈ 8 and 𝜀r ≈ 0, so
∘
2
⎧ −8 sin 30 + 8 − cos 30∘
= −0.1954 TE polarization
√
⎪
⎪ 8 sin 30∘ + 8 − cos2 30∘
Γg = ⎨
√
⎪ sin 30∘ − 8 − cos2 30∘
= −0.6868
TM polarization
√
⎪
⎩ sin 30∘ + 8 − cos2 30∘
AR =
0.6868
= 3.5155 = 10.9 dB
0.1954
5.4 Antennas over the Earth
Water
Salt water
Ice
Dry snow
Wet snow
εʹr (f )
60
40
100
20
0
Water
Salt water
Ice
Dry snow
Wet snow
80
εʺr (f )
80
60
40
20
0
10
20
30
Frequency (GHz)
40
0
50
0
10
20
30
Frequency (GHz)
40
50
(a)
Moisture content
0.1
0.2
0.3
0.4
0.5
εʹr (f )
30
20
15
0.1
0.2
0.3
0.4
0.5
10
5
10
0
Moisture content
εʺr (f )
40
0
10
20
30
Frequency (GHz)
40
0
0
50
10
20
30
Frequency (GHz)
40
50
(b)
Moisture content
0.1
0.2
0.3
0.4
0.5
εʹr (f )
15
10
10
5
0
0
Moisture content
0.1
0.2
0.3
0.4
0.5
8
εʺr (f)
20
6
4
2
10
20
30
Frequency (GHz)
40
0
0
50
10
20
30
Frequency (GHz)
40
50
(c)
Figure 5.11 Real and imaginary parts of the dielectric constants for various ground
conditions as a function of frequency. (a) Forms of water, (b) soil, and (c) vegetation.
A good example that combines multipath and reflections from lossy boundaries occurs when a wireless signal propagates in a tunnel [5]. The signal takes
many different paths as it bounces from the walls, ceiling, and floor. Material
composition of the tunnel, the tunnel shape and size, obstructions, and tunnel
bends create many opportunities for reflection, refraction, and diffraction. The
propagation path inside a tunnel may or may not have a LOS component. Consider a communications system in a tunnel with a dipole antenna that transmits
30 dBm of power to an isotropic receive antenna at the center of the tunnel cross
section [6]. The square tunnel is 3 m tall and 5 m long with dry granite walls
(𝜀r = 5, 𝜎 = 0.01 S/m). The received signal as a function of separation distance
185
5 Propagation in the Channel
Dry granite – 3 m
y = 0.1503x –1.823
1
Received power (mW)
186
0.1
0.01
0.001
0.0001
0.00001
0.000001
1
10
Distance from Tx (m)
Figure 5.12 Path loss in a mine tunnel. Source: Reprinted by permission of Ref [6]; © 2013
IEEE.
between the transmitter and receiver (d) appears in Figure 5.12. A linear fit to
the received power in Figure 5.12 leads to a path loss exponent of 1.823, so the
signal decreases according to
Pr = 0.1503x−1.823
(5.21)
The signal attenuates less than in free space. The tunnel acts like a waveguide
with lossy walls, so the signal loses energy in the walls but does not disperse as
much as in free space.
5.5 Earth Surface
Earth curvature blocks the LOS signal when the transmitter and receiver are
many kilometers apart [5]. Figure 5.13 shows two towers separated by dmax , the
dth
Horizon
Figure 5.13 The Earth
curvature limits the
maximum range of LOS
signals.
drh
ht
hr
re
re
5.5 Earth Surface
sum of the distances from the transmit (dth ) and receive antennas (drh ) to the
horizon. A LOS formula that assumes a smooth Earth with no obstructions
estimates dmax as [7]
√
√
dmax = 2ht re + 2hr re
(5.22)
where the radius of the Earth is approximately re = 6.378 × 106 m. In real life,
many natural and manmade obstructions between the transmitter and receiver
reduce dmax , so the LOS is much less than dmax .
Example
What is the maximum LOS for two microwave communication towers that are
15 m high over a smooth spherical Earth?
Solution
√
√
√
Using (5.22), it is dmax = 2ht re + 2hr re = 2 2 × 15 × 6.378 × 106 =
27.7 km.
Atmospheric density and the index of refraction decrease with height above
the Earth’s surface causing electromagnetic waves to bend or refract around the
Earth. This phenomenon lets antennas “see” beyond the horizon as shown by
the ground station communicating with a satellite in Figure 5.14. Consequently,
(5.22) underestimates dmax . An atmospheric correction to (5.22) replaces the
actual Earth by a larger imaginary Earth where the refracted electromagnetic
waves become LOS (re = rec ) [8]. The new Earth radius is approximately
rec = ke × 6.378 × 106 m
(5.23)
where the constant k e is a function of the change in the atmospheric index of
refraction (na ) with height (h) above the Earth:
ke =
1
(5.24)
dn
1 + re a
dh
Ground
station
Apparent position
Satellite
Actual position
Figure 5.14 The atmospheric index of refraction decreases with altitude, so the signal
bends around the Earth.
187
188
5 Propagation in the Channel
Atmospheric
Diffraction
re
4
r
3 e
Figure 5.15 Atmospheric refraction bends signals so that the Earth appears to have a larger
radius.
Although atmospheric conditions vary tremendously with location, time of
dn
day, and season, a reasonable average assumption of dha = 39 × 10−9 m−1 produces k e = 4/3. Thus, a quick estimate for atmospheric refraction assumes the
Earth is 1.33 times larger in radius (Figure 5.15), so the distance to horizon in
(5.22) uses (5.23) instead of the actual radius of the Earth.
Example
Repeat the previous example and take atmospheric diffraction into account.
Solution
√
√
The revised value is dmax = 2ht (4∕3)re + 2hr (4∕3)re = 31.9 km
which is over 4 km longer.
Not many surfaces are perfectly smooth as assumed with the Fresnel
equations, so, variations in ground and water height cause reflections in
directions other than predicted by Snell’s law. The impact of surface variations
depends on 𝜆, so surface variations of 1 m appear smooth at high frequency
(HF) but very rough at cell phone frequencies. Figure 5.16 shows the effect of
various degrees of surface roughness on the reflected and scattered wave. A
smooth surface has a surface height standard deviation limited by [9]
𝜎surf <
𝜆
32 cos 𝜃i
(5.25)
If (5.25) is true, then the Fresnel equations predict the direction where most
of the reflected field travels. As the surface becomes rougher, the scattered
field spreads over a wider angle and the Fresnel equations fail to accurately
predict the angular scattering of the reflected field. The dominant coherent
scattered wave obeys Snell’s law. The noncoherent part of the scattered wave
radiates isotropically. In this case, coherence implies the same polarization as
5.5 Earth Surface
Incident
wave
Scattered
wave
θi
Surface
Smooth
Very rough
Little rough
Figure 5.16 Scattering from a surface with various degrees of roughness.
the incident wave at Snell’s angle of reflection. Noncoherence refers to the coand cross-polarized components at angles other than Snell’s angle of reflection.
The reflection coefficient at Snell’s angle of reflection is given by [4]
Γ = ΓTE,TM e−(2k𝜎surf cos 𝜃i )
2
(5.26)
A perfectly smooth surface has 𝜎 surf → 0 and Γ = ΓTE or Γ = ΓTM .
Example
A 1 GHz TE wave is incident on a lake at 0 ≤ 𝜃 i ≤ 45∘ with 𝜎 surf < 2.3 cm. Plot
Γas a function of 𝜃 i and 𝜎 surf (Figure 5.17).
Solution
3 × 1010
= 30 cm, ni = 1, nt = 1.33
𝜆=
1 × 109
cos(𝜃i ) − 1.33 cos(𝜃t )
ΓTM =
cos(𝜃i ) + 1.33 cos(𝜃
(t )
)
sin 𝜃i
−1
sin 𝜃i = 1.33 sin 𝜃t ⇒ 𝜃t = sin
[
(
)] 1.33
sin 𝜃i
−1
cos(𝜃i ) − 1.33 cos sin
( )2
1.33
−4 2𝜋
𝜎 2 cos2 𝜃i
Γ=
[
(
)] e 30 surf
sin
𝜃
i
cos(𝜃i ) + 1.33 cos sin−1
1.33
–0.1
Γ
Figure 5.17 Reflection
coefficient for a 1 GHz TE
wave is incident on a lake
between 0 ≤ 𝜃 i ≤ 45∘ that
has a height variation of
𝜎 surf < 2.3 cm.
–0.15
–0.2
2
σsurf (cm)
1
0
0
10
20
30
θi (°)
40
189
190
5 Propagation in the Channel
ow y
ad ar
Sh und
bo
n
tio
lec dary
f
Re un
bo
Lit
region
y
θD
x
Diffraction
h
Shadow
region
ho
Reflection
ht
hr
dt
Screen
dr
Figure 5.18 Plane wave incident on a single edge. Dashed lines are equal phase fronts.
5.6 Diffraction
Diffraction is electromagnetic scattering from an edge. Figure 5.18 shows a
plane wave incident on a thin, perfectly conducting diffraction screen that
extends from y = 0 to − ∞. The lit region contains the incident wave, while
the shadow region does not. A shadow boundary divides these two regions.
Diffracted waves exist in both the lit and shadow regions. The reflection boundary splits the lit region in two: one with incident, diffracted, and reflected
waves and the other with only incident and diffracted waves. Reflected and
transmitted waves obey Snell’s laws, but diffraction requires substantially more
complicated calculations.
5.6.1
Fresnel Diffraction
The diffracted electric field in the shadow region due to the edge of the screen
in Figure 5.18 is given by [10]
(5.27)
Ediff = ELOS F(𝜈F )
where F(𝜈 F ) is the Fresnel integral:
𝜈
F(𝜈F ) =
∫0
e−j𝜋𝜉
2
∕2
d𝜉 = C(𝜈F ) + jS(𝜈F )
(5.28)
5.6 Diffraction
and
ELOS = LOS electric field at the receiver
√
2dt dr
𝜈F = 𝜃D
𝜆(dt + dr )
dt = obstacle distance from the transmitter
dr = obstacle distance from the receiver
(
)
(
)
ho − ht
ho − hr
−1
−1
𝜃D = tan
+ tan
dt
dr
ho = obstacle height above the ground
ht = transmit antenna height above the ground
hr = receive antenna height above the ground
𝜈
C(𝜈) = ∫0 cos(0.5𝜋𝜉 2 )d𝜉 = Fresnel cosine integral
𝜈 sin(𝜉)
S(𝜈) = ∫0
d𝜉 = Fresnel sine integral
𝜉
Note that S(∞) = C(∞) = 0.5, S(−∞) = C(−∞) = − 0.5, and S(0) = C(0) = 0.0.
Figure 5.19a has plots of the two Fresnel integrals as a function of 𝜈. As 𝜈 gets
large, the oscillations due to the diffracted field attenuate. The Cornu spiral plot
in Figure 5.19b has centers at (−0.5, −0.5) and (0.5, 0.5) which correspond to
the observation point at an infinite distance from the edge where oscillations
due to the diffracted field cease. The diffraction attenuation is [11]
1
1
0.5
0.5
S (ν)
Fresnel integral
LdiffdB
)
(√
[1 − C(𝜈) − S(𝜈)]2 + [C(𝜈) − S(𝜈)]2
= −20 log
2
0
C (ν)
S (ν)
–0.5
–1
–5
0
ν
(a)
(5.29)
0
–0.5
5
–1
–1
–0.5
0
C (ν)
0.5
1
(b)
Figure 5.19 Plots of the Fresnel integrals. (a) Cosine and sine integrals and (b) cornu spiral.
191
5 Propagation in the Channel
0
Ldiff (dB)
192
Fresnel integral
Approximation
–10
–20
–30
–5
–0.7
0
ν
5
F
Figure 5.20 Fresnel integral diffraction loss compared with approximation.
The loss in (5.29) can be estimated in dB by [12]
{
0
(√
) 𝜈F < −0.78
Ldiffdb ≈
2
6.9 + 20 log
(𝜈F − 0.1) + 1 + 𝜈F − 0.1
𝜈F ≥ −0.78
in dB
(5.30)
Example
Plot (5.29) and (5.30) on the same axis.
Solution
Figure 5.20 shows the comparison.
Since the multipath signal travels a distance greater than RLOS , the LOS signal
arrives first, then multipath signals arrive in a time proportional to the distance
traveled. In addition, each time the multipath signal reflects from a surface, it
attenuates and receives a phase shift. All single bounce multipath signals traveling the same distance have the same phase. The reflected signal has equal phase
contours that form a Fresnel ellipsoid in three dimensional space with transmit
and receive antennas at the two foci. The sum of the distances from any point
on one of the equal phase ellipses to those two foci equals RLOS + n𝜆/2 where
n is an integer. In other words, paths A and B in Figure 5.21 equal RLOS + 𝜆/2
(n = 1) while path C is 𝜆/2 longer (n = 2). The scattered field from any object
lying within the first Fresnel zone arrives at the receive antenna with a phase
difference from the LOS signal between 180∘ and 360∘ . An object in the second
Fresnel zone causes a phase shift in the scattered field between 360∘ and 540∘ .
Thus, objects close to odd numbered Fresnel zone ellipses result in the LOS
and multipath signals adding in phase. Objects close to even numbered Fresnel
zone ellipses result in the LOS and multipath signals adding out of phase.
5.6 Diffraction
3
2
1
Transmit
Receive
RLOS
A
C
B
Figure 5.21 Fresnel ellipsoids.
The radius of the nth Fresnel zone ellipse at a point that is dt from the transmitter and dr from the receiver is
√
n𝜆dt dr
(5.31)
rn =
dt + dr
Figure 5.22 shows the Fresnel zones mapped onto a diagram of the environment of a microwave link. Odd Fresnel zones experience constructive
interference, while even zones have destructive interference. Usually, up to
20% blockage in zone 1 introduces a small loss in the link [13]. More than 40%
zone 1 blockage introduces significant signal loss. This analysis breaks down
when the separation distance between antennas causes the Earth curvature to
enter zone 1.
× 10–3 Knife-edge diffraction vs. clearance distance
Field amplitude |E|/Eo
6
1
5
3
2
4
4
5
6
7
Fresnel zone map
3
ro
2
r + ro
1
0
5
10
d (λ)
15
20
r
Closest allowable clearance
2
3
4
5
Figure 5.22 The geometry of Fresnel-zone ellipsoids clearing obstacles in a microwave link.
The Fresnel zones are labeled moving from the first ellipsoid outward. Source: Reprinted by
permission of Ref. [14]; ©2009 IEEE.
193
194
5 Propagation in the Channel
Figure 5.23 A 3D Fresnel zone for a microwave link between two buildings. [15]. Image
obtained using Google Earth.
Example
A 715.4 MHz communication link between the tops of two buildings has Site
#1 located 15 m above ground at 39∘ .45′ 3.3′′ N, 105∘ 13′ 22.1′′ W. Site #2 is
located 15 m above ground at 39∘ 44′ 57.7′′ N 105∘ 13′ 21.9′′ W. These locations
were found using Google Maps. Plot the first Fresnel zone ellipsoid for this link
using 3-D Fresnel Zone found at https://www.loxcel.com/3d-fresnel-zone.
Solution
The software generates a KML file that viewed in Google Earth as shown in
Figure 5.23. The Fresnel zone plots determine whether buildings enter the first
Fresnel zone of a communications link. The antennas may have to be raised (or
trees trimmed) if obstacles lie inside the first Fresnel zone.
5.6.2
Diffraction from Multiple Obstacles
Signals diffract from edges and the Fresnel integral in (5.28) provides a relatively simple way to calculate diffraction from a single edge. Many times, the
5.6 Diffraction
Buildings
Screens
Figure 5.24 Corners, edges, and peaks of buildings are converted to screens.
Re
ce
ive
LO
Equivalent
S
screen
OS
it L
m
ns
a
Tr
Receiver
Transmitter
1 A
dt
B
2
dr
Figure 5.25 Bullington method replaces all of the screens with an equivalent screen.
signal encounters multiple edges that make the diffraction calculations more
complicated. Several methods approximate all building edges with diffraction
screens as shown in Figure 5.24. The simplest approach, the Bullington method,
replaces all of the diffraction screens with one equivalent screen (dashed arrow
in Figure 5.25) using the following steps [16]:
1. Draw straight lines from the transmitter through the top of all the screens.
The line with the steepest slope passes through the tallest screen in the direction of the receiver, such that all other screens fall below this line.
2. Draw lines from the receiver through the tops of all the screens. The line
with the steepest slope passes through the tallest screen in the direction of
the transmitter, such that all other screens fall below this line.
3. These two lines intersect at the location and height of the Bullington screen.
195
196
5 Propagation in the Channel
4. Calculate the Fresnel attenuation only for the Bullington screen while ignoring all others.
This approach works well for two closely spaced screens with no other significant obstacles.
Example
There are three knife edge obstacles between two antennas in a communications system that operates at 600 MHz:
Object
Antenna 1
Height (m)
Distance
10
0m
Obstacle A
40
7 km
Obstacle B
60
12 km
Obstacle C
30
22 km
Antenna 2
10
26 km
Find the diffraction loss using Bullington’s method.
Solution
Assume x is the distance in km from antenna 1.
30
20
x=
(26 − x)
7
4
x = 14 km
183
The height of the Bullington screen is h = 30 13×7
= 60 m. Next, the Fresnel
parameter, 𝜈 F , is found from
√ (
√ (
)
)
1
1
1
2 1
2
= 1.49
+
+
= 60
𝜈F = h
𝜆 dt dr
.5 14 000 12 000
where 𝜆 = 3 × 108 /600 × 106 = 0.5 m. Now use (5.30) to get the attenuation due
to the screen.
(√
)
(1.49 − 0.1)2 + 1 + 1.49 − 0.1 = 16.7 dB
Ldiffdb = 6.9 + 20 log10
The screen attenuation is added to the free space attenuation to get the total
attenuation.
In general, approaches to multiple screen diffraction loss divide the problem
into many one screen diffraction problems. The first step starts by calculating
the diffraction loss when the transmitter radiates from the first screen to the
second screen. Next, the diffraction loss is calculated from the first screen to
5.6 Diffraction
H2
HN – 1
H1
HN
h1
hN
H0
LOS
d1
d2
HN +1
dN
dN +1
Figure 5.26 Diagram of the Epstein–Peterson multiple screen diffraction model.
the third screen with the second screen in between. This process continues until
the last screen diffraction loss is calculated. The final loss due to all the screens
is just the sum (in dB) of all the diffraction losses due to the individual screens.
Figure 5.26 diagrams the Epstein–Peterson model [17] with the height and
distance variable labeled find the attenuation using the following steps [18]:
1. Calculate the clearance heights from the screen/antenna heights before and
after the current diffraction screen:
dn (Hn+1 − Hn−1 )
hn = Hn − Hn−1 −
(5.32)
dn + dn+1
2. Calculate the Fresnel diffraction parameter for screen n:
√
2(dn + dn+1 )
𝜈Fn = hn
𝜆dn dn+1
(5.33)
3. The total diffraction loss in dB due to all the screens is found from (5.30):
LdiffdB =
N
∑
6.9 + 20 log
(√
)
(𝜈Fn − 1)2 + 1 + 𝜈Fn − 0.1 in dB (5.34)
n=1
Example
There are three knife edge obstacles between two antennas in a communications system that operates at 600 MHz (Table 5.2):
Find the diffraction loss using the Epstein-Peterson model.
Solution
Substitute into (5.32), (5.33), and (5.34) to get the values in Table 5.2.
Adding all the losses from the individual screens results in a total loss of
197
198
5 Propagation in the Channel
Table 5.2 Given and calculated values for the Epstein–Peterson example.
Given
Object
Height (m)
Calculated
Distance
hn
𝝂 Fn
Ldiff (dB)
Antenna 1
10
0m
Obstacle A
40
7 km
11.6
0.3483
9.0
Obstacle B
60
12 km
23.5
0.5340
10.6
7.1
−0.1298
4.9
Obstacle C
30
22 km
Antenna 2
10
26 km
9.0 + 10.6 + 4.9 = 24.5 dB. The Epstein–Peterson model predicts much higher
diffraction attenuation than the Bullington model for this case.
Many other multiple screen diffraction methods exist. The Deygout method
tends to overestimate loss, especially for multiple closely spaced screens [19].
Its results are best for one dominant screen. This approach beats the Bullington
and Epstein–Peterson methods in highly obstructed paths. These approximate
methods have been surpassed by much more realistic computer models.
5.6.3
Geometrical Theory of Diffraction
Joseph Keller combined GO and Fresnel diffraction into the geometrical theory
of diffraction (GTD) [20].The total electric field equals the GO electric field
(incident, refracted, and reflected waves) and the diffracted electric field [21].
ET = EGO + Ediff
(5.35)
Figure 5.27 shows the total, GO, and diffracted amplitude and phase patterns
due to a TE plane wave incident on a perfectly conducting screen (at a 60∘
angle relative to the screen). The discontinuities in the diffracted field exactly
compensate for the discontinuities in the GO field to produce a smooth total
electric field. The diffracted field has circles of constant phase that are centered
on screen’s edge. Diffraction occurs at the discontinuity between two materials. The TM version appears in Figure 5.28. These figures show that diffraction
depends on polarization. Note the differences and similarities between the two
polarizations.
Fresnel diffraction from screens approximates the actual diffraction from
obstacles with sharp edges in the channel. The GTD combines with Fresnel
diffraction through a more rigorous formulation that includes multiple edge
diffractions and obstacle interactions [20]. Shooting and bouncing rays (SBRs)
technique practically implements ray tracing using GO and diffraction [22].
It launches rays in directions and at amplitudes proportional to the antenna
pattern in those directions. All the LOS, diffracted, refracted, and reflected
waves that reach the receiver sum to obtain a realistic received signal.
5.6 Diffraction
Total field TE polarization
(a) 4λ
360°
0 dB
–10
–20
x
–30
–40
0°
–4λ
4λ
–4λ
4λ
Geometrical optics field TE polarization
(b) 4λ
–4λ
360°
0 dB
–10
–20
x
–30
–40
0°
–4λ
4λ
–4λ
Diffracted field TE polarization
4λ
(c) 4λ
–4λ
360°
0 dB
–10
–20
x
–30
–40
0°
–4λ
4λ
y
–4λ
4λ
y
–4λ
Figure 5.27 Amplitude (left) and phase (right) plots of the (a) total, (b) geometrical optics,
and (c) diffracted fields from a screen for TE polarization (electric field perpendicular to the
plane of the plots). Source: Reproduced with permission of IEEE.
199
200
5 Propagation in the Channel
Total field TM polarization
(a)
4λ
360°
0 dB
–10
–20
x
–30
–40
0°
–4λ
4λ
(b)
4λ
–4λ
4λ
Geometrical optics field TM polarization
–4λ
360°
0 dB
–10
–20
x
–30
–40
–4λ
0°
–4λ
4λ
Diffracted field TM polarization
4λ
(c)
4λ
–4λ
360°
0 dB
–10
–20
x
–30
–40
0°
–4λ
4λ
y
–4λ
4λ
y
–4λ
Figure 5.28 Amplitude (left) and phase (right) plots of the (a) total, (b) geometrical optics,
and (c) diffracted fields from a screen for TM polarization (magnetic field perpendicular to
the plane of the plots). Reprinted by permission of Durgin [14]. © 2009 IEEE.
GTD calculates the polarization-dependent diffracted field from edges and
adds the result to the incident and reflected fields to get the total field. Elementary GTD assumes the diffracting edge consists of two infinite half planes
that meet to form a wedge as shown in Figure 5.29. The GTD diffracted electric
field for an incident plane wave with the electric field either parallel (TM) or
5.6 Diffraction
ϕ
r
rʹ
ϕʹ
Source
point
Observation
point
Figure 5.29 Geometry for application of GTD wedge diffraction.
perpendicular (TE) to the edge of a perfectly conducting wedge is [23]
Ediff =
Ei
× DTM (𝜁 , 𝜙, 𝜙′ ) ×
⏟⏟⏟
TE
Field at edge
⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟
Diffraction coefficient
where
Ei
r’
r
DTM (𝜁 , 𝜙, 𝜙′ )
𝜁
=
=
=
=
1
√
r
⏟⏟⏟
× e−jkr
⏟⏟⏟
(5.36)
Phase
Spreading factor
incident electric field
distance from source point to diffracting edge
distance from diffracting edge to field point
diffraction coefficient for a finitely conducting wedge
TE
= distance parameter defined by type of wave (planar,
spherical, etc.)
′
= incidence angle, measured from incidence face
𝜙
𝜙
= diffraction angle, measured from incidence face
More details and examples of using GTD appear in the literature [20, 23].
Assume that an FM radio station transmits 250 kW at 91.1 MHz. To reach
one coverage area, the station must rely on double diffraction over the top
of two terrain peaks of equal altitude, which can be modeled as 90∘ wedges
(Figure 5.30). The incident field at A and the two diffraction coefficients at B
and C are
At point A: Pr =
Pt Gt Gr 𝜆2
16𝜋 2 r12
201
202
5 Propagation in the Channel
ϕ2 = 225°
r2 = 500 m
150 m
TX
r1=

m
500
ϕ1=
ϕ3 = 45°
TX
00
m
ϕ1=
r3 =
500
m
150 m
RX
r2 = 500 m
5
r1=

5°
27.
ϕ2 = 270°
150 m
ϕ4 = 242.5°
°
5
72.

ϕ4 = 287.5°
ϕ3 = 90°

r3 =
5
00
m
150 m
RX
Figure 5.30 An example of FM signal propagation that must diffract over two terrain peaks.
Reprinted by permission of Durgin [14]. © 2009 IEEE.
|
|2
|
|
|
|
Pt Gt Gr 𝜆3 ||DTM (L, 𝜙2 , 𝜙1 )||
|
|
| TE
|
|
|
At point B: Pr =
3
32𝜋 r1 r2 (r1 + r2 )
|
|2 |
|2
|
||
|
|
|
|
|
Pt Gt Gr 𝜆4 ||DTM (L, 𝜙2 , 𝜙1 )|| ||DTM (L, 𝜙4 , 𝜙3 )||
|
||
|
| TE
| | TE
|
|
||
|
At receiver: Pr =
64𝜋 4 r1 r2 r3 (r1 + r2 + r3 )
For this example [14], the LOS signal at the receiver is −11.9 dBm. If the edges
are modeled as perfectly conducting screens, then the signal at the receiver for
TE is −85.5 dBm and for TM is −80.2 dBm. Modeling the edges as 90∘ wedges
results in a signal level at the receiver of −88.8 dBm for TE and −76.4 dBm
for TM.
5.7 Signal Fading
Fading occurs when the receive signal level drops below the minimum
detectable level of the receiver due to
• Path loss
• Fluctuations in the received signal power
• Fluctuations in signal phase
5.7 Signal Fading
Fading
Large scale
Path loss
Shadowing
Small scale
Multipath
delay spread
Flat
Frequency
selective
Doppler spread
Fast
Slow
Figure 5.31 Categories of fading.
• Variations in the angle of arrival of the received signal
• Reflection and diffractions from objects
• Frequency shift due to the Doppler effect
The fade margin equals the difference in dB between the received signal
strength and the receiver sensitivity. A reliable link has a high fade margin.
As an example, a 2.4 GHz, industrial, scientific, and medical (ISM) band
radio should have at least a 15 dB fade margin, and a 5 GHz ISM/Unlicensed
National Information Infrastructure (U-NII) band radio should have at least
a 10-dB fade margin [24]. Figure 5.31 breaks down the different categories of
fading that will be discussed in the following paragraphs.
Large-scale fading means that the signal amplitude slowly changes over distances greater than a wavelength. Small-scale fading, on the other hand, implies
that signal fades occur quickly over small distances on the order of a wavelength. The differences between these two types of fading are shown by signal
power plots at the bottom of Figure 5.32. Small-scale and large-scale fading add
together to get a total signal level shown in the top plot of Figure 5.32.
Two types of large-scale fading are path loss and shadowing. Path loss
attenuates a signal by 1/R𝛾 as previously discussed. Eventually, a signal fades
when R becomes large. Shadowing causes large-scale fading when a large
object (e.g. building or mountain) blocks the LOS signal, but the diffracted
signal exists and slowly decays in the shadow region.
Small-scale fading splits into two categories: multipath delay spread and
Doppler spread [25]. Flat fading means that a multipath channel has a bandwidth greater than the symbol period (Figure 5.33). All frequency components
in a flat-fading channel have the same amount of fading. Flat fading preserves
the spectral characteristics of the transmitted signal. The signal bandwidth is
203
5 Propagation in the Channel
Signal loss (dB)
0
Total signal loss
–50
–100
200
0
600
400
800
1000
Distance (m)
Small scale
Signal loss (dB)
Large scale
Signal loss (dB)
204
0
–20
–40
–60
0
200
400
600
800 1000
0
–20
–40
–60
0
200
Distance (m)
400
600
800 1000
Distance (m)
Figure 5.32 The total signal is a combination of large-scale and small-scale fading.
st (t)
Ts
0
Figure 5.33 Flat-fading
channel.
sr (t)
0
τ
Ts + τ
0
h(t,τ), τ ≪ Ts
st (f)
H (f)
sr (f)
fc
fc
fc
less than the channel bandwidth. Frequency-selective fading (time dispersion)
occurs when a multipath channel has a bandwidth that is less than the signal
bandwidth (Figure 5.34). Not all signal frequencies experience the same fading.
The signal passing through it spreads out in time and has a reduced bandwidth.
Attenuation and phase variations in the channel transfer function produce
time-varying distortion in the received signal. Fast and slow fading describe
the time rate of change in the channel transfer function and the transmitted
signal. Coherence time approximates the time duration when the channel
impulse response remains constant. Two received signals have a strong
5.7 Signal Fading
sr (t)
st (t)
Figure 5.34 Frequency
selective fading.
0
Ts
τ
0
Ts + τ
0
h(t,τ), τ > Ts
st (f )
H (f )
sr (f )
fc
fc
fc
amplitude correlation during the coherence time. Doppler spread refers to
the increased bandwidth due to the Doppler spectrum that results from
channel/transmitter/receiver motion. Fast fading has a coherence time longer
than the symbol period. Signal distortion increases as the Doppler spread
increases relative to the transmitted signal bandwidth. Slow fading has a static
channel over at least one symbol period. This means that the channel Doppler
spread is less than the baseband signal bandwidth.
5.7.1
Small-Scale Fading Models
Small-scale fading has fast variations in the signal over wavelength-size distances. Signal levels appear to be random and are very difficult to precisely predict. Experimental measurements under various practical conditions produced
data bases that have resulted in statistical models that have proved extremely
useful in system design.
5.7.1.1
Rayleigh Fading
Consider the case of many signals arriving in a region via different paths. Each
signal has a different amplitude and phase at the receiver due to the differences
in path lengths as well as the diversity of reflections, diffractions, and refractions encountered. If the 2 GHz signals in Table 5.3 converge over a 6𝜆 × 6𝜆 area,
they form an interference pattern. Figure 5.35a,b show the real and imaginary
parts of the total electric field due to these multipath signals. Converting the
real and imaginary parts of the total field to amplitude and phase produces the
plots in Figure 5.35c,d. These plots clearly indicate that the signal over this small
area dramatically oscillates. A mobile wireless user passing through this area of
interference encounters large signal variations and fading.
205
206
5 Propagation in the Channel
Table 5.3 Ten 2 GHz multipath signals at a receiver.
Signals at the receiver
0.9835 e−jk(x cos 0
∘ + y sin 0∘ ) j0∘
e
−jk(x cos 71.2 ∘ + y sin 71.2∘ ) j341.6∘
0.7736 e
e
0.9863e−jk(x cos 310.7
∘ + y sin 310.7∘ ) j283.1∘
e
−jk(x cos 354.0 ∘ + y sin 354.0∘ ) j311.9∘
0.8574e
0.8489e−jk(x cos 59.0
e
∘ + y sin 59.0∘ ) j62.3∘
e
0.6080e−jk(x cos 215.0
∘ + y sin 215.0∘ ) j27.0∘
e
−jk(x cos 3.2 ∘ + y sin 3.2∘ ) j216.3∘
0.9881e
e
0.5031e−jk(x cos 139.2
0.6265e−jk(x cos 15.9
∘ + y sin 139.2∘ ) j60.5∘
e
∘ + y sin 15.9∘ ) j264.0∘
0.7174e−jk(x cos 344.4
e
∘ + y sin 344.4∘ ) j147.0∘
e
Example
Use MATLAB to make histogram plots of the field values in Figure 5.35.
Solution
When these plots were made, the values of the total electric field, ET , were
saved over a 6𝜆 × 6𝜆 area. The MATLAB command hist(real(ET), 30), where ET
contains the sum of all the signals in Table 5.3, results in the plots in Figure 5.36.
Gaussian curves nicely fit the real and imaginary histograms but not the amplitude and phase histograms (Table 5.4).
Note that the real and imaginary field values in Figure 5.36a,b resemble Gaussian probability density functions (PDFs). Figure 5.37 has two Gaussian PDF
plots: one with 𝜇 = 0 and a standard deviation of 𝜎 = 1 (also known as the standard normal distribution) and the other with 𝜇 = 2 and 𝜎 = 0.5. The standard
deviation describes how close the points cluster about the mean.
The PDFs in Figure 5.36c,d cannot be Gaussian, because
1. The amplitude is always greater than or equal to zero, so the PDF must begin
at x = 0.
2. One phase is equally likely as another phase, so the PDF must be uniform.
It turns out that when the real and imaginary parts of a random variable have
Gaussian PDFs with a standard deviation of 𝜎, then the phase is a uniform PDF
while the amplitude is a Rayleigh PDF takes the form of
x −
PDF(x ∣ 𝜎) = 2 e
𝜎
(
x2
2𝜎 2
)
(5.37)
10
10
5
5
Imag (ET)
Re (ET)
5.7 Signal Fading
0
–5
0
–5
–10
–10
80
80
60
60
y (cm)
y (cm)
40
20
0 0
20
40
80
60
40
x (cm)
20
0
0
20
(a)
60
40
x (cm)
80
(b)
10
∠ ET (°)
|ET|
100
5
0
–100
0
80
80
60
60
y (cm) 40
20
y (cm) 40
0
0
20
60
40
x (cm)
80
20
0
(c)
0
20
60
40
x (cm)
(d)
Figure 5.35 Total electric field due to the ten multipath signals in Table 5.3 and no LOS
signal. (a) Real, (b) imaginary, (c) amplitude, and (d) phase.
Table 5.4 Statistics of the field data.
Real
Imaginary
Amplitude
Phase
Mean
0.00
0.00
2.00
−0.08
Variance
2.73
2.73
1.05
3.35
80
207
3000
Number in 6λ × 6λ area
Number in 6λ × 6λ area
5 Propagation in the Channel
2000
1000
–4
–2
0
2
4
–4
–2
0
(a)
(b)
1500
1000
500
1
1000
Imag (ET)
2000
0
2000
Re (ET)
2500
0
3000
0
–6
6
Number in 6λ × 6λ area
Number in 6λ × 6λ area
0
–6
2
3
4
5
6
4
2
6
1500
1000
500
0
–200
–100
0
|ET|
∠ ET (°)
(c)
(d)
100
200
Figure 5.36 Histograms of the sum of the fields in Table 5.3 with plots of the PDFs found
from the data statistics. (a) Real, (b) imaginary, (c) amplitude, and (d) phase.
0.8
μ = 2, σ = 0.5
0.6
PDF (x)
208
μ = 0, σ = 1
0.4
0.2
0
–4
0
x
–2
2
4
Figure 5.37 Two examples of Gaussian PDFs.
with its cumulative density function (CDF) given by
x
CDF(x ∣ 𝜎) =
∫0
y − y22
2
2
e 2𝜎 dy = 1 − e−x ∕(2𝜎 )
𝜎2
(5.38)
where 𝜎 is the mode or the location of the maximum value of the PDF.
Figure 5.38 contains three examples of the Rayleigh PDF for 𝜎 = 0.5, 1.0,
and 2.0. Some statistics of a Rayleigh distribution appear in Table 5.5. Adding
5.7 Signal Fading
1.5
σ = 0.5
σ = 1.0
σ = 2.0
PDF (x)
1
0.5
0
0
1
2
x
3
4
Figure 5.38 Rayleigh PDFs.
Table 5.5 Rayleigh PDF statistics.
√
𝜎 π∕2
Mean
Median
2𝜎 2
√
𝜎 2 ln(2)
Variance
0.429𝜎 2
Mean square value
many multipath signals with no LOS signal results in a received signal with an
amplitude that is Rayleigh distributed and a phase that is uniformly distributed.
Example
The average power in a Rayleigh faded signal is given by the mean square value
in Table 5.5. What is the probability that the signal falls 10 dB below its mean
square value?
Solution
Use(the CDF )
in (5.38) to obtain:
2
1
= 1 − e−1∕10 = 0.0952
p 2𝜎x 2 < 10
5.7.1.2
Rician Fading
The previous example lacks a dominant or LOS signal. A Rayleigh model
works well for modeling a smart phone in a big city where the base station
hides behind buildings. Systems like satellite television have a dominant LOS
signal with smaller multipath signals, the Rayleigh model does not work. What
happens if the first signal in Table 5.3 increases to an amplitude of 6 while the
other signals remain the same? Figure 5.39 has plots of the total field consisting
209
10
10
5
5
Imag (ET)
Re (ET)
5 Propagation in the Channel
0
–5
0
–5
–10
–10
80
80
60
60
y (cm) 40
20
y (cm) 40
20
0
0
20
60
40
x (cm)
80
0
0
20
60
40
x (cm)
80
(b)
(a)
10
∠ ET (°)
100
|ET|
210
5
0
–100
0
80
80
60
60
y (cm) 40
y (cm) 40
20
0
0
20
60
40
x (cm)
(c)
80
20
0
0
20
60
40
x (cm)
80
(d)
Figure 5.39 Total electric field due to one LOS and nine multipath signals. (a) Real, (b)
imaginary, (c) amplitude, and (d) phase
of the LOS signal and 9 multipath signals (the bottom 9 signals in Table 5.3).
Figure 5.39a,b show the real and imaginary parts of the total electric field due
to these multipath and LOS signals. Converting the total field to amplitude
and phase results in the plots in Figure 5.39c,d. These plots of the real,
imaginary, and amplitude exhibits fading like those in Figure 5.35. The phase
in Figure 5.39d exhibits a periodicity corresponding to the dominant LOS
signal.
2000
Number in 6λ × 6λ area
Number in 6λ × 6λ area
5.7 Signal Fading
1500
1000
500
0
–10
–5
0
5
10
2000
1500
1000
500
0
–10
–5
Re (ET)
2000
1000
0
2
4
5
10
200
400
(b)
Number in 6λ × 6λ area
Number in 6λ × 6λ area
(a)
3000
0
0
Imag (ET)
6
8
10
1200
1000
800
600
400
200
0
–400
–200
0
|ET|
∠ ET (°)
(c)
(d)
Figure 5.40 Histograms of the sum of nine multipath signals and a LOS signal. (a) Real, (b)
imaginary, (c) amplitude, and (d) phase.
Example
Use MATLAB to make histogram plots of the field values in Figure 5.39.
Solution
ET now contains the sum of the bottom 9 signals in Table 5.3 plus a LOS signal
with an amplitude of 6, phase of 0, and arriving at 0∘ . Figure 5.40 plots the
resulting histograms.
The real and imaginary field values in Figure 5.40 do not look like those
in Figure 5.36. They have two peaks and cannot be modeled with Gaussian
PDFs. The phase histogram looks uniform like the histogram in Figure 5.36d,
but the amplitude histogram differs from that in Figure 5.36c. The LOS signal
dominates all the other signals, so the mass of the PDF moves closer to the
amplitude of the LOS signal. The new PDF in Figure 5.40c takes the form of a
Rician PDF [26]
(
)
(x2 +A2LOS )
ALOS x
x − 2𝜎MP
2
(5.39)
I0
PDF(x ∣ 𝜎MP , ALOS ) = 2 e
2
𝜎MP
𝜎MP
where
ALOS
2
2𝜎MP
I0 (𝜉) =
1
2𝜋
= amplitude of the LOS signal
= average power of the non-LOS multipath signals
2𝜋 𝜉 cos 𝜃
∫0 e
d𝜃 = zeroth order modified Bessel function of the first
kind
211
5 Propagation in the Channel
0.6
Figure 5.41 Two
examples of Rician PDFs.
ALOS = 0.5, σ = 1
ALOS = 2, σ = 1
PDF (x)
212
0.4
0.2
0
1
0
2
x
3
4
A graph of two Rician PDFs, both with 𝜎 MP = 1, but one having ALOS = 0.5 and
the other with ALOS = 2 appear in Figure 5.41. As ALOS gets small, the Rician PDF
approaches a Rayleigh PDF and as ALOS gets large, the Rician PDF approaches
a Gaussian PDF.
The Rice factor equals the ratio of the power in the LOS signal to the average
power of the sum of the multipath signals [27]:
Kr =
A2LOS
2
2𝜎MP
(5.40)
If there is no LOS signal, then ALOS = 0 and (5.39) becomes a Rayleigh PDF. A
strong LOS signal converts (5.39) to an approximate Gaussian distribution with
a mean of ALOS . In Figure 5.41, K r = 2 for the solid curve and K r = 0.125 for the
dashed curve. More advanced Rician models include two or more dominant
signals, such as the LOS and a ground reflection.
5.7.2
Approximate Channel Models
The Rician and Rayleigh models apply to situations that have many multipath
signals with and without a LOS signal. A number of refined models applicable to
specific types of channels have been developed [28]. Channel models based on
experimental measurements have interpolated between measurement points
to estimate the signal loss due to commonly encountered wireless environments over specific frequency ranges. Of all the reported measurements, the
Okumura report data became a standard [29]. This report presented graphs
of the median field strength in a channel as a function of base station height,
frequency, and vehicular station antenna height. The report contains data collected from 150 to 1920 MHz, at distances between 1 and 100 km, and with base
station antenna heights from 30 to 200 m as well as over various types of urban
areas in and around Tokyo.
5.7 Signal Fading
213
Table 5.6 Parameters for the Hata model.
Propagation area
Metropolitan
Small to
medium
cities
a(hr )
8.29[log(1.54hr )]2 − 1.1 fc ≤ 200 MHz
3.2[log(11.75hr )]2 − 4.97 fc ≥ 400 MHz
(1.1 log f c − 0.7)hr − 1.56 log f c + 0.8
C
0
0
Suburban
−2[log(f c /28)]2 − 5.4
Rural
−4.78(logf c )2 + 18.33 log f c − 40.94
Hata derived equations that fit the curves in the Okumura report [30]. The
general Hata attenuation model curve fit is
LHata = A + B log d + C
(5.41)
where d is the separation distance in km between the transmitter and receiver.
The values of A, B, and C depend upon the propagation environment. Hata’s
model gives A and B as
A = 69.55 + 26.16 log fc − 13.82 log ht − a(hr )
B = 44.9 − 6.55 log ht
(5.42)
where the transmitter height (ht ) and receiver height (hr ) are in meters and the
carrier frequency (f c ) is in MHz. The values of a and C given in Table 5.6 depend
upon the propagation environment (column 1). Many other propagation models exist for different environments and parameters.
Example
A base station operates at 900 MHz at a height 30 m above the ground. If the
receiver is 1 km away at a height of 2 m above ground, then find the path loss
using the Hata model for (a) large city, (b) small city, and (c) rural area.
Solution
LHata = 69.55 + 26.16log10 fc − 13.82 log ht − a(hr ) + (44.9 − 6.55 log ht )
log d + C = 126.42 − a(hr ) + C
(a) 125.37 dB
(b) 125.13 dB
(c) 96.62 dB
214
5 Propagation in the Channel
Indoor models predicting signal attenuation based on measurements also
exist. For example the Motley–Keenan indoor model attenuation factor due
to propagation in and through buildings is given by [31]
( )
d
(5.43)
+ Lwall + Lfloor in dB
Lindoor = L0 + 10𝛾 log
d0
where
L0
d
= path loss at d0 (dB)
= separation distance (m) between the transmitter and receiver
(d > 1 m)
𝛾
= power loss exponent
= wall attenuation (dB)
Lwall
= floor attenuation (dB)
Lfloor
If the reference distance is do = 1 m and assuming free space propagation,
then L0 = 20 log 10 f MHz − 28 where f MHz is in MHz. Table 5.7 lists measured
values of 𝛾 for various types of buildings at different frequencies. Table 5.8
lists wall attenuation through commonly used materials in walls. Table 5.9
lists some floor attenuation factors in different scenarios and for different
frequencies. Both Lwall and Lfloor in (5.43) include all walls and floors that the
signal passes through.
Example
A 900 MHz signal enters a building through a 13-mm thick window. It passes
through one concrete wall that is 102 mm thick and one floor. The signal travels
a total of 10 m in the building. Estimate the attenuation.
Solution
Let d0 = 1 m and substitute the values from
) tables into (5.43).
( the
10
Lindoor = 20 log(900) − 28 + 10(3.3) log 1 + 2 + 12 + 9 = 87.1 dB
5.7.3
Large-Scale Fading
Obstacles in a channel block the LOS signal and create shadows (Figure 5.18).
Shadow regions contain the diffracted signal whose amplitude is considerably
lower than the LOS signal. Obstacles in a cell change the ideal circular coverage
into an amoeba-like coverage area shown in Figure 5.42. Mobile users passing
behind an obstacle experience a fade in the obstacle’s shadow.
The attenuated signal in a shadow appears random with a log-normal PDF.
The PDF is based on the ratio of the received power to the transmit power in
dB:
(𝜒dB −𝜇𝜒dB )
−
2
1
e 2𝜎𝜒dB
(5.44)
PDFA = √
2𝜋𝜎𝜒dB
5.7 Signal Fading
Table 5.7 Power loss exponents, 𝛾, for indoor transmission loss calculation in dB [32].
𝜸
Frequency (GHz)
Residential
Office
Commercial
Factory
0.8
—
2.25
—
—
0.9
—
3.3
2.0
—
1.25
—
3.2
2.2
—
1.9
2.8
3.0
2.2
—
2.1
—
2.55
2.0
2.11
2.2
—
2.07
—
—
2.4
2.8
3.0
—
—
2.625
—
4.4
—
3.3
3.5
—
2.7
—
—
4
—
2.8
2.2
—
4.7
—
1.98
—
—
5.2
3.0 2.8
3.1
—
—
5.8
—
2.4
—
—
26
—
1.95
—
—
28
—
1.84 2.99
2.76 1.79 2.48
—
37
—
1.56
—
—
38
—
2.03 2.96
1.86 2.59
—
51–57
—
1.5
—
—
60
—
2.2
1.7
—
67–73
—
1.9
—
—
70
—
2.2
—
—
300
—
2.0
—
—
where
𝜒 = Pr ∕Pt
𝜒dB = 10 log(Pr ∕Pt )
𝜇𝜒dB = mean of 𝜒dB in dB
𝜎𝜒dB = standard deviation of 𝜒dB in dB ∶ typically 4 ≤ 𝜎𝜒dB ≤ 13 dB
Figure 5.43 plots 𝜒 dB vs. separation distance in dB. Each point on the graph
represents the receive signal sample for a constant transmitter power. A linear
interpolation of the scatter plot has a slope of −10𝛾 log d where 𝛾 is the power
loss exponent. Points follow a Gaussian PDF with a mean of −10𝛾 log d and a
standard deviation of 𝜎𝜒dB . The mean of 𝜒 dB depends on path loss and obstacle
215
216
5 Propagation in the Channel
Table 5.8 Approximate attenuation through a single wall at 900 MHz [32].
Material
Thickness
Lwall (dB)
Glass
6
0.8
Glass
13
2
Lumber
76
2.8
Brick
89
3.5
Brick
267
5
Concrete
102
12
Concrete
203
23
Reinforced concrete
203
27
Masonry block
610
28
Concrete
305
35
Table 5.9 Floor penetration loss factors (Lfloor ) for Nfloor floors [32].
Lfloor (dB)
Frequency (GHz)
Residential
Office
Commercial
0.9
—
9 (N floor = 1)
19 (N floor = 2)
24 (N floor = 3)
—
1.8–2
4 N floor
15 + 4 (N floor − 1)
6 + 3 (N floor − 1)
2.4
10a) (apartment)
14
—
5 (house)
3.5
—
18 (N floor = 1)
26 (N floor = 2)
—
5.2
13a) (apartment)
7b) (house)
16 (N floor = 1)
—
5.8
—
22 (N floor = 1)
28 (N floor = 2)
—
a) Per concrete wall.
b) Wooden mortar.
properties. The total fade margin for a wireless system is the large-scale fade
margin plus the small-scale fade margin in dB.
The decorrelation distance in a shadow, X c , corresponds to the maximum
distance in which two signals remain correlated. Two signal samples become
decorrelated when their separation distance causes the signal covariance to
5.7 Signal Fading
Figure 5.42 A free space cell has a circular
boundary while a shadowed cell has an
irregular cell boundary.
Cell with shadowing
Circular cell
Antenna
Obstacles
Figure 5.43 Log normal
fading with propagation
loss.
–1
0γ
lo
d)
Received signal
χdB
g(
σ χdB
σ χdB
log (d)
drop 1/e below its maximum. A simplified model for the signal covariance is
given by [33]
cov(d) = 𝜎𝜒2dB e−d∕Xc
(5.45)
where d is the separation distance between two points in the shadow. The
decorrelation distance approximately equals the width of the obstacle.
5.7.4
Channel Ray-Tracing Models
Very accurate computer models use numerical techniques to solve Maxwell’s
equations. These models have long run times for realistic environments, work
best at low frequencies, and only work well for small regions. Consequently,
approximations and statistics supplement these approaches. One popular
high-frequency approximation, ray tracing, finds practical use in modeling
217
218
5 Propagation in the Channel
Figure 5.44 Ray tracing in an urban canyon. Source: Reproduced with permission of
Remcom.
complicated scenarios. A transmitter emits a ray that propagates into a region
of interest, such as a city. The ray attenuates, reflects, and diffracts as it takes
diverse paths and propagates away from the transmitter. The SBR technique
launches a bundle of rays with amplitudes weighted by the transmit antenna
pattern [22]. GO and diffraction determine the direction and scattering of a
ray. Figure 5.44 illustrates an example of the SBR technique used in Wireless
Insight [34] that models an antenna transmitting from a tall building. After
positioning transmit and receive antennas, the algorithm only calculates the
rays that propagate to the receiver. When the amplitude of a ray drops below
a user-defined threshold, it is ignored. SBR in a complex environment is
computationally intensive, because a large number of rays must be launched
and tracked.
SBR has many applications to wireless system modeling. The example in
Figure 5.45 models Wi-Fi coverage in a house due to a transmitter on the
first floor. This model helps determine the optimum router location in order
to obtain coverage throughout the house. The scenario in Figure 5.46 shows
an antenna on top of a building that transmits into another building. Some
regions have good coverage while others do not. This type of model helps
determine the location of an indoor signal boosting system that provides
uniform coverage inside the building.
5.8 Doppler Effects
Figure 5.45 Wi-Fi coverage in a house with the router on the first floor. Source: Reproduced
with permission of Remcom.
Figure 5.46 Predicting interior coverage from an exterior transmitter. Source: Reproduced
with permission of Remcom.
5.8 Doppler Effects
When the transmitter and/or receiver moves the transmitted signal experiences
a Doppler shift that translates the signal to a lower or higher frequency. Most
people associate Doppler shift with sound. A train approaching the listener has
a higher pitch than a train going away from a listener. Wireless systems experience Doppler in low-orbit satellite and cellular communication systems due to
the velocities of the transmitter and receiver. Doppler causes the carrier frequency to shift higher when the transmitter approaches, and lower when it
recedes. The Doppler shifted frequency is calculated using:
)
(
ktr
kc − vr ⋅ ̂
fc
(5.46)
fD =
kc − vt ⋅ ̂
ktr
219
220
5 Propagation in the Channel
where
= receiver velocity vector
vr
vt
= transmitter velocity vector
̂
ktr
= unit propagation vector from transmitter to receiver
The difference between the Doppler shifts in signals coming from the same
transmitter via two different paths is the Doppler spread. Frequency-shifted
signals from multiple paths interfere with each other and cause fading.
Example
A car traveling at 25 m/s receives a 2 GHz signal from a stationary transmitter.
Find the received signal frequency when the car travels (a) directly toward the
transmitter, (b) directly away from the transmitter, and (c) in a circle with the
transmitter as the center.
Solution
8
= 0.15 m
The wavelength is 𝜆 = 3×10
2×109
Using (5.46):
(
)
x ⋅ (−̂
x)∕0.15
2 × 109 − 25̂
25
2 × 109 = 2 × 109 +
(a) fD =
0.15
2 × 109
= 2.0 000 001 667 GHz
(
)
x ⋅ (̂
x)∕0.15
2 × 109 − 25̂
25
(b) fD =
2 × 109 = 2 × 109 −
9
0.15
2 × 10
= 1.999 999 833 GHz
(
)
2 × 109 − 25̂
x ⋅ (̂
y)∕0.15
(c) fD =
2 × 109 = 0
2 × 109
This frequency shift looks small, but if the maximum and minimum Doppler
shifted signals are added together, the resultant signal experiences fading.
Doppler shift occurs when the transmitter and receiver move together or
apart. No Doppler shift occurs when the separation distance remains constant.
Fading also takes place when the multipath signals and the LOS signal have
different Doppler shifts. For example if the transmitting car in Figure 5.47
moves at a velocity vt and the receiving car moves at a velocity vr , then the
LOS signal (̂
ktr ) and the multipath signal (̂
kmp ) have different Doppler shifts
according to (5.46). These two signals add together to produce fading.
Doppler fading occurs when adding two sinusoids of different frequencies – one due to LOS (f c + f D1 ) and one due to multipath (f c + f D2 ). The
resulting signal takes the form
[
]
[
]
(
)
(
)
cos 2𝜋(fc + fD1 )t + cos 2𝜋(fc + fD2 )t = 2 cos 2𝜋flow t cos 2𝜋fhigh t
(5.47)
5.8 Doppler Effects
Figure 5.47 Doppler effect
due to moving and
stationary platforms and
obstacles.
K̂ mp
V
t
K̂ tr
V
r
where
fD1 − fD2
2
2fc + fD1 + fD2
fhigh =
2
The low-frequency signal modulates the high-frequency signal in (5.47).
Thus, the envelope of the modulated signal goes to zero or fades every
1/|f D1 − f D2 |. A beat frequency is the difference between two closely spaced
frequencies, |f D1 − f D2 |. When tuning a guitar, the guitarist adjusts the string
tension in order to minimize beat frequency between the plucked string and a
tuning fork.
flow =
Example
A 1 GHz LOS signal arrives at the receiver with a multipath signal at 1.075 GHz.
Plot the resultant signal that shows the beat period in the time domain.
Solution
A time vector was defined between 0 and 20 ns in MATLAB. The LOS and
multipath signals were summed and graphed: cos(2𝜋 × 109 t) + cos(2𝜋 ×
1.075 × 109 t) for 0 ≤ t ≤ 20 ns. The beat period is given by 1/|1.075 × 109 −
1.0 × 109 | = 13.3 ns (Figure 5.48).
The Doppler spectrum extends from fc − fDmax to fc + fDmax , where the Doppler
spread for a stationary transmitter and moving receiver (vr ) is defined as the
221
5 Propagation in the Channel
Figure 5.48 Beat frequency.
Beat period
2
Amplitude (V)
222
1
0
–1
–2
5
0
10
t (ns)
15
20
maximum Doppler frequency
v
fDmax = r
(5.48)
𝜆
Doppler shift due to any velocity translates into frequencies within the
Doppler spectrum. If the baseband signal has a bandwidth greater than 2fDmax ,
then the Doppler spread has little impact on the received signal, and it is a
slow fading channel. A high Doppler spread produces fast fading and requires
a receiver that tolerates a fade within one symbol. As long as the symbol rate
exceeds fDmax , then Doppler in the channel can be ignored.
Coherence time, T c is the inverse of Doppler spread [35].
Tc =
1
fDmax
(5.49)
A low Doppler spread produces slow fading (T c ≫ T s ), so the impulse
response remains invariant over the coherence time. Two received signals have
a strong amplitude correlation if they are less than T c apart. If T s > T c , then the
channel changes during the transmission of the baseband symbol and results
in a distorted symbol recovered by the receiver. Two signals arriving more
than T c apart encounter different channel effects. Fading due to the Doppler
shift is called time selective fading. (T c < T s ). A more conservative estimate of
T c is given by
Tc =
9
16𝜋fDmax
(5.50)
The coherence time in (5.49) is longer than in (5.50). A common compromise
between the two is given by
Tc =
0.423
fDmax
(5.51)
5.9 Fade Margin
S (f)
Figure 5.49 Doppler spectrum.
fc – fD max
fc
fc + fD max
f
Example
A car is traveling at 25 m/s receives a 2 GHz signal from a stationary transmitter.
Find the coherence time using Eqs. (a) (5.49), (b) (5.50), and (c) (5.51).
Solution
25
fDmax =
= 166.67 Hz
0.15
1
1
=
= 6 ms
(a) Tc =
fDmax
166.67
9
(b) Tc =
= 1.07 ms
16π(166.67)
0.423
= 2.54 ms
(c) Tc =
166.67
If equal amplitude sinusoidal multipath signals at frequency f c arrive with
equal probability from all angles surrounding a receiver that is moving at a
velocity vr , then the Doppler spectrum takes the form [36]
1
S(f ) = √
(5.52)
2
𝜋 fD − (f − fc )2
max
which is graphed in Figure 5.49. This plot shows the carrier-wave spectrum
extending from fc − fDmax to fc + fDmax . The low frequency occurs when the
receiver moves away from the transmitter, while the high frequency occurs
when the receiver moves toward the transmitter. All other motion produces
frequencies between these extremes.
5.9 Fade Margin
The level crossing rate (LCR) is the expected number of times that the fading
signal amplitude goes above an established threshold (V thresh ). Rayleigh fading
223
224
5 Propagation in the Channel
has an LCR of [10]
√
2
LCR = 2𝜋fDmax 𝜌T e−𝜌
(5.53)
where 𝜌T is the threshold voltage normalized to the rms signal level (V rms ) written as
𝜌T = Vthresh ∕Vrms
(5.54)
The LCR is proportional to the mobile receiver speed.
The average fade duration (AFD) estimates the average length of time that
the signal spends below the threshold (signal not received). For Rayleigh fading
and isotropic scattering, the AFD below a level 𝜌T is [10]
e𝜌T − 1
√
𝜌T fDmax 2𝜋
2
AFD =
(5.55)
For a particular threshold value, the product of the AFD and the LCR is a
constant.
2
AFD × LCR = 1 − e−𝜌
(5.56)
Example
A mobile receiver traveling at 60 mph operates at 900 MHz with fDmax = 88 Hz.
If the threshold is 0 dB, then find the LCR and AFD.
Solution
LCR =
√
√
2
2𝜋fDmax 𝜌T e−𝜌T = 2𝜋(88)(1)e−1 = 81 fades∕s
AFD =
e1 − 1
e𝜌T − 1
=
= 7.8 ms
√
√
𝜌T fDmax 2𝜋
(1)(88) 2𝜋
2
2
As a check, use (5.56): AFD × LCR = 0.0078 × 81 = 1 − e−𝜌T = 0.632
5.10 Atmospheric Propagation
The atmosphere has five distinct layers of decreasing density with increasing
height above the ground. Layer thickness depends on latitude, season, and time
of day. Approximate layer thicknesses at mid-latitude (roughly 30–60∘ ) are [37]
•
•
•
•
•
Troposphere: up to 14.5 km
Stratosphere: up to 50 km
Mesosphere: up to 85 km
Thermosphere: up to 600 km
Exosphere: up to 10 000 km
5.10 Atmospheric Propagation
The troposphere and thermosphere substantially impact RF propagation
compared to the other layers, because the troposphere contains almost
all the water vapor as well as precipitation. This moisture attenuates electromagnetic waves due to the complex permittivity of water in its various
forms (Figure 5.11). At a much greater height, the ionosphere (part of the
thermosphere) either reflects waves at lower frequencies or attenuates and
depolarizes waves at all other frequencies.
Wireless propagation usually occurs in the troposphere where most of the
atmospheric moisture resides. Water has a high dielectric constant, so the more
water in the troposphere, the more the RF signal attenuates. The rain attenuation coefficient, Lrain , is a function of the number and size of the raindrops
∞
Lrain =
∫0
 ()()d
(5.57)
Marshall and Palmer [38] proposed a simple exponential rain drop size distribution (DSD):
 () = Ndr e−𝜌r 
(5.58)
where  () is the number of drops having an equivalent spherical diameter
 (mm) by unit volume, for a diameter interval of 1 mm. They proposed that
N dr = 8 × 103 mm−1 m−3 and 𝜌r = 4.1Rain −0.21 mm−1 with Rain in mm/h. This
distribution is widely used due to its simplicity and good fit for mid-latitude
DSDs which have low to moderate rain intensity. () is the attenuation
cross-section that corresponds to the attenuation due to a single spherical
raindrop. An approximation in the Rayleigh region ( ≪ 𝜆) is
() ∝ 3 ∕𝜆
(5.59)
The attenuation calculated in (5.57) reduces the received power density
as shown by the loss term that multiplies the Friis transmission formula in
Figure 5.50. Figure 5.51 plots the attenuation vs. frequency for several rainfall
rates. The attenuation becomes quite significant at higher frequencies.
Water vapor and atmospheric gases also attenuate signals. Figure 5.52 has
plots of the attenuation curves associated with water vapor and oxygen. Tropospheric attenuation due to water and oxygen is negligible at low frequencies.
Single
raindrop
Rainfall
PtGte – αrainr
4 πr 2
PtGt
Transmit
Figure 5.50 Raindrops attenuate signals passing through.
Receive
225
5 Propagation in the Channel
Attenuation (dB/km)
6
0.5 mm/h
2.5 mm/h
10 mm/h
20 mm/h
4
2
0
0
20
10
30
40
Frequency (GHz)
Figure 5.51 Rainfall attenuation as a function of rainfall rate and frequency.
H2O
40
20
O2
10
Figure 5.52 Atmospheric gas attenuation:
solid line is water vapor and dashed line is
oxygen [39].
O2
Attenuation (dB/km)
226
1
0.4
0.2
0.1
H2O
0.04
0.02
0.01
0.002
10
20
50
100
200 350
f (GHz)
Attenuation due to water vapor steadily increases with frequency while oxygen
drops off above 100 GHz (except for the resonances). Water has resonances at
22 and 183 GHz while oxygen resonates at 60 and 120 GHz. Wireless communications systems avoid the frequency bands around these spikes.
Normally, the warmest air lies next to the Earth. Temperature usually
decreases with height above the Earth. Atmospheric waveguides, called ducts,
form when
1. An inversion causes cool air to lie next to the Earth with a layer of warm air
above it.
2. A layer of cool air gets trapped between two layers of warmer air.
5.10 Atmospheric Propagation
Figure 5.53 Depolarization
of an electromagnetic
wave.
Excrossx̂
z
Excox̂
+
Eycrossŷ
Eycoŷ
Exx̂
Medium
Eyŷ
The layer of cool air has a higher dielectric constant than the warm air, so
RF signals at certain frequencies in this atmospheric waveguide encounter low
propagation loss over great distances [40]. The signal between a transmitter
and receiver in the same duct decays much less than in free space. Ducting
mostly occurs over large bodies of water at ultra high frequency (UHF) and
microwave frequencies [3]. Elevated ducts occur up to about 1500 m above the
surface when the layers of the atmosphere trap the wave from both above and
below. Surface ducting is more common and of more practical use, since wireless systems tend to reside there. Ducting conditions occur more frequently
in some locations, but their randomness precludes reliable use as a communications channel. Ducting may cause unintentional interference to a distant
communication system that is normally too far away to receive the signal.
An electromagnetic wave depolarizes as it passes through a medium. Depolarization means that a plane wave traveling in the z-direction has part of its
x-component converted into a y-component and/or vice versa. An electromagnetic wave entering the depolarization medium with an x-component (Ex ) and
a y-component (Ey ) exits the medium with co-polarized (Exco , Eyco ) and cross
polarized (Excross , Eycross ) components (Figure 5.53). Examples of depolarization
media include the atmosphere, vegetation, and the ground. Cross polarization
discrimination (XPD) measures the amount of depolarization [41]
XPD = 20 log
Eco
Ecross
(5.60)
√
2
2
where Eco = Exco
+ Eyco
is the co-polarized electric field amplitude and
√
2
2
Ecross = Excross
+ Eycross
is the cross polarized electric field amplitude.
The other region of the atmosphere that significantly impacts radiowaves
is the ionosphere. Radiation from the Sun ionizes atoms and releases free
electrons in the ionosphere. These free electrons attenuate, refract, and reflect
electromagnetic waves depending upon frequency (Figure 5.54). Electron
density starts growing at approximately 30 km above the Earth, but only
noticeably impacts radio signals from about 60 to 90 km. Skywaves are radio
signals in the low MHz range that reflect from the ionosphere. These HF
227
228
5 Propagation in the Channel
F2
F1
E
D
High
frequency
dis Skip
tan
ce
Medium
frequency
Sun
Low
frequency
F
Multiple
skip
Figure 5.54 The ionosphere affects the propagation distance.
signals propagate around the globe by bouncing between the ionosphere and
the Earth multiple times. The distance along the Earth between the transmitter
and the point on the ground where the signal reflected from the ionosphere
returns to the Earth is called the skip distance and is approximated by [42]
√
(5.61)
dskip = 2 2re h′
′
where h is the virtual height of the ionospheric layer (height where the
transmitted and received rays appear to cross). The skip zone is a region
transmitter and the skip distance where the signal is too weak to detect.
The ionosphere has several layers as shown in Figure 5.55 (night is on the left
half and day on the right half ). The D layer lies closest to the Earth at 50–80 km
during the day and disappears at night [43]. The electrons quickly recombine
at this height, because the relatively high air density provides many opportunities for finding an ion in need of an electron. The Earth blocks solar radiation
at night causing the electron levels to fall, and the D layer effectively disappears. Consequently, low frequency signals that reflect from the D layer cannot
reach higher layers in the ionosphere except at night when the region disappears. Signals passing through the D region attenuate as the inverse square of
the frequency.
The E layer lies above D at 100–125 km above the Earth [43]. Unlike the D
layer, it refracts more than attenuates signals passing through. At night, ion
5.10 Atmospheric Propagation
F2 Layer 300–400 km
F Layer 250–300 km
F1 Layer 200–300 km
E Layer 100–125 km
D Layer 50–80 km
0 10 102 103 104 105 106 107
electrons/cm3
Mesosphere < 85 km
0 10 102 103 104 105 106 107
electrons/cm3
Stratosphere < 50 km
Troposphere < 14.5 km
Night
Day
Figure 5.55 Diagram of the ionosphere with nighttime on the left and daytime on the right.
The dashed curve represents the typical electron density as a function of height [45].
density in the E layer dramatically decreases as shown by the dashed line plots in
Figure 5.55. A typical electron lifetime in the E region is 20 seconds. The residual nighttime ionization at the bottom of the E region causes some attenuation
of signals in the lower portions of the HF part of the radio communications
spectrum. Signals reflected from the E layer pass back through the D region
and get attenuated before returning to Earth. The maximum skip distance for
the E region is around 2500 km.
Meteors, electrical storms, auroral activity, and upper atmosphere winds
create highly ionized pockets in the E layer that last for a few hours. This
sporadic E layer horizontally extends from a few meters to several hundred
kilometers while being only a few kilometers thick [44]. When this layer
initially forms, it impacts low frequencies; then as time passes, it impacts
increasingly higher frequencies until it reaches a peak before decreasing as
it disperses. Sporadic E reflects higher-frequency signals from a few minutes
to a few hours and lower-frequency signals for much longer. The maximum
skip distance is around 2000 km. Sporadic E causes some very high frequency
(VHF) signals to propagate further while preventing HF signals from reaching
the higher F regions, thus reducing their propagation distance. In temperate
regions, sporadic E occurs mainly in the summer. Generally, the higher VHF
frequencies are only affected in the middle of the sporadic E season. Auroral
sporadic E usually occurs in the morning at the polar regions and have little
variation with the seasons. Sporadic E exists primarily in the daytime near the
equator with little seasonal variation. In temperate climates, two main peaks
occur: one at noon and the other at 19:00.
229
230
5 Propagation in the Channel
The F layer lies above the E layer and facilitates long-distance communication
at HF frequencies below. Its altitude depends on the time of day, the season, and
the amount of solar activity. During the day it splits into two layers called F1
and F2. In the summer, the F1 layer is at 300 km, with the F2 layer at 400 km or
more. In the winter, these layers significantly drop in altitude. At night, the F
layer is between 250 and 300 km. A typical electron lifetime in the F1 region is
1 minute and the F2 region 20 minutes. The maximum skip distance for the F2
region is around 5000 km.
Spread F contains electron density irregularities caused by ionospheric
storms in the F2 region below. It lasts from minutes to hours and horizontally
extends up to hundreds of kilometers. HF signals experience fading when
passing through the spread F region. It usually occurs in high latitudes at night
but can occur during the day as well. Spread F is not common at mid-latitudes.
The number of electrons as a function of height above the Earth determines
the reflection and attenuation properties of the ionospheric layers. The Total
Electron Content (TEC) is the number of electrons present along a path
between a radio transmitter and receiver [46]. One TEC Unit corresponds to
1016 electrons/m2 . The TEC depends on the local time, latitude, longitude,
season, geomagnetic conditions, solar cycle and activity, and troposphere
conditions. A high TEC reduces the position measurement accuracy of
satellite navigation systems. The Global Positioning System (GPS), the US
part of Global Navigation Satellite System (GNSS), uses an empirical model
of the ionosphere, the Klobuchar model, to calculate and remove part of
the positioning error caused by the ionosphere when single frequency GPS
receivers are used.
The ionospheric electron density depends upon the solar activity. Measuring
the ionizing radiation from the sun provides an accurate prediction of ionospheric behavior. The solar radio flux at 10.7 cm or 2800 MHz (F10.7 index)
is measured in Solar Flux Units (SFUs). This index indicates the amount of
solar activity, because emissions at 2800 MHz correlate well with the sunspot
number as well as ultraviolet and extreme ultraviolet emissions that impact
the ionosphere. Measurements of the F10.7 index started in Ottawa, Canada
in 1947 and still continue at the Penticton Radio Observatory in British
Columbia. Figure 5.56 lists the predicted sunspot number and radio flux values
with expected ranges on 8 August 2015. The Wolf number (Swolf ) [47] measures
the number of sunspots and groups of sunspots observed on the surface of the
sun. A model that relates Swolf to SFU is given by [48]
2
SFU = 63.7 + 0.728Swolf + 0.00089Swolf
(5.62)
As of 2015, the Brussels International Sunspot Number used in Figure 5.56
(SB ) supplanted Swolf due to improved observations [49]. They are related by
Swolf = 0.6SB
(5.63)
5.10 Atmospheric Propagation
Figure 5.56 Sunspot numbers (SB ) and radio flux [50]. Source: Reproduced with permission
of NOAA.
Geomagnetic activity also affects the ionosphere. The solar wind (charged
particles emanating from the sun) changes the shape of the Earth’s magnetic
field, which in turn disturbs the charged particles in the ionosphere [51]. The
convective motion of charged, molten iron inside the Earth generates the magnetosphere (right side of Figure 5.57). The solar wind constantly bombards the
sun-facing side of the Earth’s magnetic field and causes it to compress until it
extends from 6 to 10 times the radius of the Earth. The magnetosphere on the
night side of the Earth has a long tail that fluctuates but extends up to hundreds
of Earth radii (beyond the moon).
Magnetic observatories around the globe record the largest magnetic change
(maximum minus minimum) that occurred in a three-hour period. Two indices
describe the observed level of geomagnetic activity [53]:
1. K p : The K index lies between 0 and 9 with higher numbers corresponding to
higher geomagnetic activity. The estimated planetary K index (K p ) is the
mean standardized K-index from 13 geomagnetic observatories between
44∘ and 60∘ northern or southern geomagnetic latitude. Figure 5.58 plots
the estimated Kp over a three-day period.
2. Ap : The three-hour Ap (equivalent range) index is derived from the K p index.
Since the A is derived from the K index then averaged over the period of
a day. Like the K index, values are averaged around the globe to give the
planetary Ap index. Table 5.10 equates the K p and Ap indexes to geomagnetic
activity.
231
5 Propagation in the Channel
Figure 5.57 Image of Earth’s magnetosphere [51]. Source: NASA [52] https://images-assets
.nasa.gov/image/0201490/0201490~orig.jpg.
Estimated Planetary K index (3 hour data)
Begin: 2015 Aug 19 0000 UTC
9
8
K>4
7
K=4
6
5
4
K<4
Kp index
232
3
2
1
0
Aug 19
Aug 20
Universal time
Updated 2015 Aug 21 21:00:22 UTC
Aug 21
Aug 22
NOAA/SWPC Boulder, CO USA
Figure 5.58 Estimated K p index over a three day period [52]. Source: Reproduced with
permission of NOAA.
5.10 Atmospheric Propagation
Table 5.10 Measures of geomagnetic activity [53].
Ap index
K p index
Activity
0
0
Quiet
4
1
Quiet
7
2
Unsettled
15
3
Unsettled
27
4
Active
48
5
Minor storm
80
6
Major storm
132
7
Severe storm
208
8
Very major storm
400
9
Very major storm
The critical frequency (f crit ) is the lowest frequency that passes through the
ionosphere. An ionosonde measures the critical frequency by transmitting a
signal vertically from the ground and recording the return at the same location.
The signal time delay indicates the height of the ionospheric layers. The critical
frequency for the E layer is from 1 to 4 MHz and for the F2 layer from 2 to
13 MHz [54]. The lowest critical frequency occurs at night during the lowest
sunspot years. The highest critical frequency occurs at daytime in high solar
activity years. Periods of high solar activity drive the critical frequency to as
high as 20 MHz for brief periods. The critical frequency relates to the electron
density in an ionospheric layer by
√
(5.64)
fcrit = 9 Ne
where N e is the electron density in number of electrons per m3 .
The maximum usable frequency (MUF) is the highest frequency that reflects
from the ionosphere when the signal is incident at an angle 𝜃
fcrit
(5.65)
cos 𝜃
where 𝜃 = 0 is vertical. The skip distance dramatically increases as the frequency
approaches the MUF.
The Lowest Usable Frequency (LUF) is the lowest frequency that results in
satisfactory reception. The LUF increases with increasing solar activity. Sometimes the LUF exceeds the MUF between a transmitter and receiver, because
the highest possible frequency that propagates through the ionosphere to the
receiver gets absorbed [54].
The Frequency of Optimum Traffic (FOT) is the highest frequency that
is usable 90% of the days in a month for a specific propagation path [55].
MUF =
233
234
5 Propagation in the Channel
Sometimes FOT is referred to as Optimum Working Frequency (OWF). FOT
is estimated to be 85% of the median monthly MUF. To maintain continuous
operation, the wireless system operates at a higher frequency during the day
and a lower frequency at night.
Near Vertical Incidence Skywave (NVIS) propagation works below the critical frequency (between 3 and 10 MHz) by reflecting a signal from the ionosphere at an angle less than 15∘ from normal [56]. NVIS covers an area about
200 km in radius that has no intermediate man-made infrastructure. It works
well when mountains, buildings, and trees block LOS signals. The desired coverage area and solar activity dictate the optimum operating frequency. It is ideal
for disaster relief communication, communication in developing regions, and
military applications.
Problems
5.1
Find the received power if 100 W is transmitted 40 km through a 40 dB
antenna. The receive antenna has a gain of 23 dB and the frequency is
5 GHz.
5.2
A handset radiates 1 W through a 3 dB linear polarized antenna at
2.4 GHz. Assume that the base station is polarized matched to the
antenna 1 km away. Estimate the size of the base station aperture that is
needed to receive the signal for a receiver with a sensitivity of −100 dBm.
5.3
A plane wave is incident from medium 1 with n1 = 1.0 onto two layers:
medium 2 with n2 = 1.5 and medium 3 with n1 = 2.0. Find the transmission angle into medium 3.
5.4
A wave is incident from a region with ni = 1.0 to a region with nt = 1.5.
For 0 ≤ 𝜃 i ≤ 90∘ plot the TE and TM (a) reflection coefficients and (b)
transmission coefficients.
5.5
A wave is incident from a region with ni = 1.5 to a region with nt = 1.0.
For 0 ≤ 𝜃 i ≤ 90∘ plot the magnitude of the TE and TM reflection coefficients.
5.6
On the same graph, plot the reflection coefficient vs. angle for nt = 1.0
and a transmission region with (a) n1 = 1.5 and (b) n2 = 2.5.
5.7
Glass has an 𝜀r = 2.25. Calculate the percentages of reflected and transmitted powers for a 2.4 GHz signal at normal incident on a large planar
sheet of glass.
Problems
5.8
A 100 ns pulse at f c = 2.4 GHz takes two paths. One path is 100 m and
the other is 105 m. How long is the received pulse?
5.9
A 100 MHz LOS communications system has transmitting and receiving
antennas separated by 15 km over a flat Earth. If the transmitting antenna
is 20 m above ground, what is the optimal height of the receive antenna?
5.10
An LOS link at 50 MHz has a transmit antenna 20 m above ground. The
receive antenna is 15 km away. Find the height of the receive antenna that
will maximize signal reception for TE polarization.
5.11
A 1 GHz RHCP wave is incident on flat ground covered in deep wet snow
at 𝜓 g = 20∘ . What is the axial ratio of the reflected wave?
5.12
If antennas are placed on 20 m towers what should their separation be
on a spherical Earth when (a) no atmospheric refraction and (b) atmospheric refraction.
5.13
A 1 GHz wave is incident on a lake that has a height variation of
𝜎 surf = 2 cm. Find the reflection coefficient as a function of angle.
5.14
Plot a three-dimensional cornu spiral S(𝜈) vs. C(𝜈) vs. 𝜈.
5.15
Estimate the diffraction loss at 900 MHz for
(a) dt = 100 m, dr = 100 m, ht = 5 m, hr = 5 m, ho = 10 m
(b) dt = 100 m, dr = 100 m, ht = 5 m, hr = 5 m, ho = 5.1 m
(c) dt = 10 m, dr = 190 m, ht = 5 m, hr = 5 m, ho = 5.1 m
(d) dt = 10 m, dr = 190 m, ht = 5 m, hr = 1 m, ho = 10 m
(e) dt = 10 m, dr = 10 m, ht = 1 m, hr = 1 m, ho = 10 m
5.16
Find the radii of the first three Fresnel zones when
(a) dt = 100 m, dr = 100 m, f = 500 MHz
(b) dt = 100 m, dr = 100 m, f = 5 GHz
(c) dt = 150 m, dr = 50 m, f = 500 MHz
(d) dt = 150 m, dr = 50 m, f = 5 GHz
5.17
There are two knife-edge obstacles between two antennas in a communications system:
235
236
5 Propagation in the Channel
Object
Height (m)
Distance (m)
Antenna 1
20
0
Obstacle A
30
100
Obstacle B
20.75
225
Antenna 2
1.5
250
Find the diffraction loss using Bullington’s method at 2.4 GHz.
5.18
There are four knife-edge obstacles between two antennas in a communications system:
Object
Height (m)
Distance (km)
Antenna 1
20
0
Obstacle A
40
1
Obstacle B
50
1.5
Obstacle C
60
2
Obstacle D
10
2.2
Antenna 2
10
3
Find the diffraction loss using Bullington’s method at 800 MHz.
5.19
There are three knife-edge obstacles between two antennas in a communications system:
Object
Height (m)
Distance (m)
Antenna 1
10
0
Obstacle A
20
100
Obstacle B
25
200
Obstacle C
30
250
Antenna 2
2
300
Find the diffraction loss using Bullington’s method at 5 GHz.
5.20
Repeat Problem 17 using the Epstein–Peterson method.
5.21
Repeat Problem 19 using the Epstein–Peterson method.
5.22
For Rayleigh fading, what is the probability that the received signal power
is at least (a) 20, (b) 6, and (c) 3 dB below the mean power?
5.23
Determine the fade margin in a Rayleigh channel if the received power
falls below the receiver sensitivity 1% of the time.
Problems
5.24
Show that for K r = 0 the Rician PDF becomes a Rayleigh PDF.
5.25
A base station operates at 2.4 GHz at a height 10 m above the ground. If
the receiver is 100 m away at a height of 1.5 m above ground, then find
the path loss using the Hata model for (a) large city, (b) small city, (c)
suburban, and (d) rural area.
5.26
A base station operates at 800 MHz at a height 20 m above the ground.
If the receiver is 500 m away at a height of 10 m above ground, then find
the path loss using the Hata model for (a) large city, (b) small city, (c)
suburban, and (d) rural area.
5.27
A 2400 MHz signal enters an office building through a brick wall that is
267 mm thick. The signal travels a total of 20 m in the building. Estimate
the attenuation.
5.28
A 2.4 GHz signal enters a house through a brick wall that is 89 mm thick
and passes through one floor. The signal travels a total of 5 m in the
house. Estimate the attenuation.
5.29
A wireless communication system operates at two frequencies: (a)
900 MHz and (b) 1800 MHz. Find the maximum Doppler spread if the
transmitter is stationary and you are on a train traveling at 200 km/h.
Give an estimate of the coherence time.
5.30
Find the LCR for 𝜌T = 0.1 and a maximum Doppler frequency of 30 Hz.
5.31
Find the AFD for a maximum Doppler frequency of 200 Hz and (a)
𝜌T = 1, (b) 𝜌T = 0.1, and (c) 𝜌T = 0.001.
5.32
Compute LCR for an f Dmax = 20 Hz, f c = 900 MHz, and 𝜌T = 1. What is
the maximum velocity of the vehicle?
5.33
The U.S. Geological Survey (USGS) categorizes rain fall as follows:
violent shower Rain > 50 mm/h, heavy shower 50 ≥ Rain > 10 mm/h,
moderate shower 10 ≥ Rain > 2 mm/h, slight shower Rain < 2 mm/h. Plot
attenuation vs. frequency for 1, 5, 25, and 50 mm/h.
5.34
Plot rainfall attenuation vs. rainfall rate 50 ≥ Rain > 1 for f = 10, 20, 30,
and 40 GHz.
5.35
Find the sunspot number, 10.7-cm radio flux, and the estimated planetary K index for the day
237
238
5 Propagation in the Channel
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https://www.electronics-notes.com/articles/antennas-propagation/
ionospheric/solar-indices-flux-a-ap-k-kp.php (accessed 29 May 2018).
http://www.sidc.be/silso/datafiles-old (accessed 30 May 2018).
http://www.swpc.noaa.gov/products/predicted-sunspot-number-and-radioflux (accessed 17 July 2018).
https://images-assets.nasa.gov/image/0201490/0201490~orig.jpg (accessed
29 May 2018).
https://www.swpc.noaa.gov/products/planetary-k-index (accessed 21 August
2015).
Poole, I. (2002). Understanding solar indices. QST Magazine, pp. 38–40.
American Radio Relay League (2015). ARRL Handbook, 93e.
Kishore, K. (2009). Antenna and Wave Propagation. New Delhi: I.K. International Publishing House.
Witvliet, B.A. and Alsina-Pagès, R.M. (2017). Radio communication via near
vertical incidence skywave propagation: an overview. Telecommun. Syst. 66:
295. https://doi.org/10.1007/s11235-017-0287-2.
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6
Satellite Communications
Satellite communications refers to a communication link that involves a
satellite. Most early satellite communication operated at C band with relatively
low power and low antenna gain. The Earth stations for these satellites
transmitted several kilowatts of power through large reflector antennas that
were many meters in diameter. Future applications require higher frequencies
and smaller ground stations. Table 6.1 lists the satellite frequency bands
currently in use. Direct Broadcast Satellite (DBS) provide consumers with a
direct-to-home Satellite TV link. Fixed Satellite Service (FSS) connects two
ground stations via a satellite.
6.1 Early Development of Satellite Communications
Arthur C. Clarke initiated the idea of providing worldwide communications
using satellites in 1945 [2]. His idea did not materialize until the first rocket
launches in the late 1950s, though. On 4 October 1957, the Soviet Union started
the space race by launching a 58-cm diameter sphere weighing 84 kg called
Sputnik into orbit [3]. For three weeks, it transmitted rapid beeps at 20.007
and 40.002 MHz. Sputnik circled the Earth 1440 times before burning up in the
atmosphere on 4 January 1958. In 1960, the United States responded with Echo,
NASA’s first communications satellite. It was a 30-m diameter balloon made of
Mylar (Figure 6.1) [4]. A ground station transmitted circularly polarized signals
at 960 MHz and 2.39 GHz to the Echo satellite that reflected the signal to a different ground station. One of the ground stations, Bell Laboratory in Holmdel,
NJ, used the newly invented Hogg antenna (Chapter 4) shown in Figure 6.2. The
goals of Echo were to demonstrate long-distance voice communications, study
propagation effects, try different satellite tracking techniques, and determine
whether a passive satellite fits into telecommunications [5].
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
242
6 Satellite Communications
Table 6.1 Satellite frequencies (GHz) [1].
Frequency band
Downlink (DL)
Uplink (UL)
C
3.700–4.200
5.925–6.425
X (Military)
7.250–7.745
7.900–8.395
Ku (Europe)
FSS: 10.700–11.700
DBS: 11.700–12.500
Telecom:
12.500–12.750
FSS & Telecom:
14.000–14.800
DBS: 17.300–18.100
Ku (USA)
FSS: 11.700–12.200
DBS: 12.200–12.700
FSS: 14.000–14.500
DBS: 17.300–17.800
Ka
18–31
EHF
30–300
V
36–51.4
Source: www.nasa.gov.
Figure 6.1 Echo satellite [4]. Source: Courtesy of NASA. https://www.nasa.gov/multimedia/
imagegallery/image_feature_559.html.
6.1 Early Development of Satellite Communications
Figure 6.2 Ground station Hogg horn antenna in Holmdel, NJ [4]. Source: Courtesy of NASA.
www.nasa.gov.
In 1961, OSCAR 1 (Orbiting Satellites Carrying Amateur Radio), became
the world’s first nongovernment satellite [6]. Some amateur radio operators
built the satellite for less than US$100. It transmitted the Morse code message
“hi–hi” at 144.983 MHz for nearly 20 days, using a 60 cm monopole. The small
power supply with no solar cell for recharging limited the satellite’s lifespan.
Thousands of radio operators in 28 different countries listened to the signal.
Today, the Radio Amateur Satellite Corporation (AMSAT) operates several
satellites for use by amateur radio operators worldwide.
In 1962, the American Telephone and Telegraph Company’s (AT&T)
Telstar 1 [7, 8] became the first active communications satellite as well as the
first commercial payload in space. It was 0.9 m in diameter and weighed 77 kg.
Telstar made possible the first transatlantic television transmission (Andover
Earth Station, Maine to the Pleumeur-Bodou Telecom Center, Brittany,
France). The Andover Earth Station used an even larger version of the Hogg
horn antenna shown in Figure 6.2. A solar array with a 15-W battery back-up
powered the spin-stabilized satellite. The uplink (UL) had eight channels at
6 GHz, while the downlink (DL) had eight channels at 4 GHz [9]. It received
telemetry commands through a VHF quadrafilar helical antenna. Telstar failed
when passing through the Van Allen Belt, because the electronics were not
sufficiently radiation hardened. As a result, the satellite was deactivated in
1963. During its short life, Telstar demonstrated the feasibility of practical
satellite communications, made advances in satellite tracking, and provided
critical satellite design information on the Van Allen radiation belts. Telstar
instigated a debate in the United States on whether communications satellites
should be operated and controlled by the private sector or government.
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6 Satellite Communications
Figure 6.3 Syncom IV-3 Satellite. Source: Courtesy of NASA. www.nasa.gov.
Syncom (synchronous communication satellite) launched into a geosynchronous orbit in 1963 [10, 11]. A satellite in geosynchronous orbit circles the
Earth once every sidereal day (23 hours, 56 minutes). An observer on Earth
sees a geosynchronous satellite at the same time and in the same place in the
sky every day. Syncom had a frequency-translation transponder capable of
one two-way telephone or 16 one-way teletype channels. A slotted dipole with
2 dB of gain that operated at two frequencies near 7.36 GHz received ground
signals. The satellite retransmitted the received signals through another slotted
dipole at 1.815 GHz. Four monopole antennas extended from the cylindrical
satellite for telemetry and command. The satellite reached the desired orbit
but never functioned.
Later in 1963, Syncom 2 became the first successful geosynchronous communications satellite [12]. Syncom 2 hosted the first telephone conversation
between heads of government via satellite (USA President Kennedy and Nigerian Prime Minister Balewa). In addition, the satellite enabled several test television transmissions from Fort Dix, New Jersey to a ground station in Andover,
Maine. Other Syncoms followed eventually leading to the INTELSAT satellites.
Figure 6.3 is a photograph of a later model, Syncom IV-3, taken from the Space
Shuttle.
NASA launched the first commercial communications satellite in 1965,
INTELSAT I or Early Bird, into geosynchronous orbit [10]. It enabled 240
6.2 Satellite Orbits
Figure 6.4 Model of the TDRS satellite at the Smithsonian National Air and Space Museum.
two-way transatlantic telephone channels compared to Syncom I’s single
channel [13]. INTELSAT I through IV were spin-stabilized. Technological
advances allowed INTELSAT V to use the more advanced three-axis stabilization. The early INTELSATs operated at C band. Ku band was added later. The
C band link had a 500 MHz bandwidth that was divided into 12 subbands of
40 MHz each. The Ku subbands were at least 80 MHz wide.
In the 1980s, NASA deployed the Tracking and Data Relay Satellite (TDRS)
System in geostationary orbit (Figure 6.4) [14]. These satellites relay data from
user satellites in lower orbits to ground stations that are not in the LOS. Each
TDRS satellite has two 5-m dish antennas (S and K bands), and a 30-element
S-band phased array to communicate with the user satellites. A Ku-band
antenna communicates with the ground station. The multi-beam phased array
has 20 beams that link up to 20 different satellites.
6.2 Satellite Orbits
The mission defines the type of orbit for a satellite. An orbit determines a satellite’s speed as well as its distance from and position above the Earth. These
factors play an important role in determining the link budget.
245
6 Satellite Communications
Apogee
Figure 6.5 Satellite orbits and
their inclination.
Polar
Circular
Inclined elliptical
Perigee
Orbital period (h)
5
LEO
Earth
Orbital speed (km/s)
10 000
10
MEO
30 000
20 000
GEO
15
20 000
Orbit radius (km)
Figure 6.6 Orbit height,
speed, and period [15].
20
10 000
30 000
Inclination describes the orbit tilt angle relative to the equatorial plane.
Figure 6.5 shows a circular equatorial orbit (0∘ ) as well as an inclined elliptical
orbit and the extreme polar orbit (90∘ ). A polar orbit means the satellite orbits
Earth in a plane containing both poles. The point farthest from the Earth on
an elliptical orbit is called apogee, while the closest point is perigee.
Orbits are also classified by their distance from Earth (Figure 6.6): low Earth
orbit (LEO), medium Earth orbit (MEO), and geosynchronous Earth orbit
(GEO). Any satellite orbiting the Earth beyond GEO is in a high Earth orbit
(HEO). A satellite’s speed and orbital period determine its height above the
40 000
246
Height above sea level (km)
69.4
55.6
41.7
3.1
6.2 Satellite Orbits
Earth. Some space missions require satellites to reach an escape velocity of at
least 11.2 km/s to leave Earth orbit.
In 1945, Arthur C. Clarke talked of launching a satellite into geostationary
orbit [16], “A rocket which can reach a speed of 8 km/s parallel to the earth’s
surface would continue to circle it forever in a closed orbit; it would become an
‘artificial satellite’.” He went on to say that [16] “An ‘artificial satellite’ at the correct distance from the earth would make one revolution every 24 hours; i.e. it
would remain stationary above the same spot and would be within optical range
of nearly half the earth’s surface.” In honor of his brilliant ideas, a geostationary
orbit is called the Clarke orbit.
Geosynchronous satellites travel at about 3 km/s in an orbit at 35 800 km
above the Earth and circle the Earth once a day. A geostationary orbit is a
geosynchronous orbit in the equatorial plane. A single satellite in geostationary orbit “sees” approximately 1/3 of the Earth surface between ±77∘ latitude,
so three equally spaced satellites in geostationary orbit cover the entire Earth
outside of the polar regions. The advantages of geostationary satellites are:
•
•
•
•
Ground tracking not needed
No communications handoff needed between satellites
Three satellites provide complete Earth coverage below 77∘ latitude
Very little Doppler shift
The disadvantages include
Long transmission latencies (travel time from sender to receiver)
Received signal very weak
Poor coverage above 77∘ latitude
Expensive launch
A geostationary satellite with an antenna that has a 17.3∘ beamwidth covers approximately 1/3 of the Earth surface outside of the polar regions [17].
Multiple satellites with narrow beams are more typically used for Earth coverage. Ground stations on the equator at the same longitude as the satellite
experience a 239 ms minimum time delay – a long time in our nanosecond
world. Most ground stations are not directly below the satellite, so the cross
range distance to the satellite increases the free space loss which means the
signal is even weaker and the time delay greater. In addition, the signal passes
through more of the atmosphere which further absorbs and depolarizes the signal. Atmospheric effects limit the elevation angle for C band to 5∘ above the
horizon and for Ku band to 10∘ above the horizon.
Typically a satellite lasts about 13 years in orbit. The gravity of the sun and
moon push a satellite north or south of its geostationary orbit. East and west
deviations result from orbital velocity and altitude errors as well as the Earth not
being a perfect sphere. Without an orbital correction, a satellite moves about
0.85∘ per year in the north–south direction [17]. Orbital station-keeping moves
•
•
•
•
247
248
6 Satellite Communications
a satellite back to its intended orbit by firing thrusters. Orbital corrections use
precious fuel, and a satellite’s life expectancy depends on the amount of fuel
stored. Making corrections only when the satellite shifts ±3∘ extends a satellite’s
life by about three years. The satellite needs to save enough fuel at the end of its
life in order to safely deorbit.
The sun acts like an RF noise source that moves along a trajectory across
the sky. Twice a year during the equinoxes, the sun crosses a line between the
geostationary satellite and ground station and causes fading at the ground station for several minutes a day over a period of several days. During this time,
the increased noise degrades the carrier-to-noise ratio (C/N) of the weak signals from the satellite. The solar fading angle is the angle (measured from the
ground station antenna) between the satellite and the Sun at the time when signal degradation begins or ends. The sun outage start and end dates depend on
the geographical location of the ground station. Small antennas with very wide
beamwidths are out of commission for up to half an hour. High gain antennas
(most satellite antennas) typically have the link disrupted for only a few minutes. Complicated algorithms exist to precisely calculate when and how long
the sun fade lasts.
Simple geometry leads to an approximation for the sun fade time [18]. From
the Earth, the sun has an apparent diameter of 0.53∘ . If the ground antenna has
a 3 dB beamwidth of 𝜃 3dB in degrees, and the sun moves across the sky at a rate
of 15∘ per hour, then the maximum sun fade time in minutes is given by
Tsunfade = 4(𝜃3dB + 0.48) minutes
(6.1)
∘
The sun’s declination changes by 0.4 per day, so the maximum number of
sun fade days is given by
𝜃 + 0.48∘
Nsunfade = 3dB ∘
(6.2)
0.4
Example
What is the diameter of a parabolic reflector ground station operating at 4 GHz
that has N sunfade = 2.
Solution
𝜃3dB + 0.48∘
⇒ 𝜃3dB = 2(0.4) − 0.48 = 0.32∘
0.4∘
3 × 108
𝜆=
= 0.075 m
4 × 109
𝜃3dB = 0.32∘ = 29.2∘ ∕(ra ∕𝜆) = 29.2∘ ∕(ra ∕0.075) ⇒ ra = 6.84 m
Nsunfade = 2 =
diameter = 13.69 m
6.2 Satellite Orbits
Figure 6.7 Front and side views of the Airlink antenna. Source: Reprinted by permission of
Haupt 2010 [19]. © 2010, IEEE.
Geostationary satellites are important for weather and communications,
because ground satellite dishes do not have to track the satellites. Inmarsat, a
British satellite telecommunications company, offers worldwide communications services via 12 geostationary satellites. The AIRLINK conformal array
on airplanes provides in-flight communications via the Inmarsat geostationary
satellites [19]. Figure 6.7 shows the antenna array that operates between 1530
and 1559 MHz on receive and 1626.5 and 1660.5 MHz on transmit with a gain
in excess of 12 dB. The array of rectangular microstrip elements arranged in a
triangular grid scans ±60∘ in azimuth and elevation.
LEO satellites travel at 7.8 km/s or less depending upon the distance from the
Earth. At 160–2000 km above the Earth, they avoid the atmospheric drag and
lie below the high radiation levels of the inner Van Allen radiation belt. Space
debris resides in low orbits, so satellite survival depends on keeping track of
space debris. NASA estimates that 500 000 objects between 1 and 10 cm orbit
Earth with average impact speeds greater than 22 000 mph [20]. LEO satellites
have orbital inclinations anywhere from equatorial to polar. They are relatively
cheap to launch, have low latency, and low free space attenuation.
The Iridium constellation has 66 active LEO satellites plus several spares
in low-Earth polar orbit at 781 km above ground and an inclination of 86.4∘
in order to provide voice and data communications [21]. One orbit takes
100.5 minutes. The satellite constellation has six orbital planes spaced 30∘
apart, with 11 active satellites in each plane (Figure 6.8). Satellites control their
orbital altitudes to within 10 m and position errors to within 15 km.
The Iridium-NEXT satellite has an L-Band phased array with 168 transmit
and receive modules that produce 48 transmit/receive beams [23]. It employs
a Time-Division Duplex architecture that allocates different time slots for UL
®
249
250
6 Satellite Communications
Figure 6.8 Iridium
constellation. Source:
Reprinted by permission of
Leopold and Miller [22]. ©
1993, IEEE.
and DL in the 1616–1626.5 MHz band. The array has a 4700 km footprint on
Earth. Iridium has a 2.4-kbs voice communications data rate, a 64 kbs data
rate for L-Band Handset Data Services and Short Burst Data, and a capability of 512 kbit/s to 1.5 Mbps link for high data rate applications. There is also a
Ka-Band 8 Mbps link.
Iridium-NEXT satellites also have two 20 GHz ULs and 30 GHz DLs
connecting to a terrestrial gateway [23]. Four 23.18–23.38 GHz crosslinks
allow adjacent satellites in the same orbital plane and in adjacent planes to
route data in order to provide worldwide coverage. Crosslink communications
occurs at 12.5 Mbps, in half duplex mode. Two fixed antennas enable in-plane
communications and steerable antennas lock onto satellites in neighboring planes. Telemetry and command occur over the 20/30 GHz links with
omni-directional antennas on the satellite.
MEO satellites lie between 2000 and 35 800 km and orbit the Earth between 2
and 24 hours. The Global Positioning System (GPS) in a MEO orbits the Earth
in 12 hours at 20 200 km [24]. Figure 6.9 shows several versions of the GPS satellites. A constellation of 24 GPS III satellites appears in Figure 6.10. The satellites
in the GPS constellation surround the Earth in six equally spaced orbital planes.
Each plane contains four “slots” occupied by baseline satellites. This 24-slot
arrangement ensures users see at least four satellites from virtually any point
on the planet. GPS consists of three segments: the space segment, the control
6.2 Satellite Orbits
GPS IIR-M
GPS IIF
GPS IIA launched 1990–1997
GPS IIR launched 1997–2004
GPS IIR-M launched 2005–2009
GPS IIF launched 2010–2016
GPS III launched 2017-
GPS IIA
GPS III
Figure 6.9 GPS satellites. Source: Courtesy of United States Government [25].
www.usa.gov/government-works.
Figure 6.10 GPS satellite
orbits. One GPS III satellite
shown in orbit. Source:
Courtesy of United States
Government [25].
www.usa.gov/
government-works.
segment, and the user segment. The US Air Force develops, maintains, and
operates the space and control segments. Figure 6.11 displays the locations of
the Earth stations for ground control and tracking of GPS satellites. Glonass
(navigation satellites for Russian defense) and Galileo (navigation system used
by European Union) satellite constellations have MEO orbits as well.
251
252
6 Satellite Communications
GPS Control Segment
Greenland
Alaska
Schriever AFB
United Kingdom
New Hampshire
Vandenberg AFB Colorado
USNO Washington
California
Cape Canaveral
Florida
Hawaii
South Korea
Bahrain
Guam
Kwajalein
Ecuador
Ascension
Uruguay
Diego Garcia
South Africa
Master Control Station
Alternate Master Control Station
Ground Antenna
AFSCN Remote Tracking Station
Air Force Monitor Station
NGA Monitor Station
Australia New
Zealand
Figure 6.11 Earth stations for ground control and tracking of GPS satellites [25]. Source:
Courtesy of United States Government. www.usa.gov/government-works.
In June 2011, the Air Force successfully completed a GPS constellation
expansion called the “Expandable 24” configuration [24]. Three of the 24 slots
were expanded, and six satellites were repositioned, so that three of the extra
satellites became part of the constellation baseline. As a result, GPS now
effectively operates as a 27-slot constellation with improved coverage in most
parts of the world. As of 30 June 2017, there were a total of 31 operational
satellites in the GPS constellation, not including the decommissioned, on-orbit
spares. The GPS constellation contains old and new satellites.
Satellites send timing and position information to a receiver that calculates
the distance to the satellite. The receiver accurately calculates latitude, longitude, and altitude when it detects signals from four or more satellites. Determining the latitude and longitude of the receiver only requires three satellites.
Figure 6.12 has a block diagram of the signal generation in a GPS satellite.
The GPS signal contains three types of data [26]:
1. Pseudo-random code identifies the transmitting satellite.
2. Ephemeris data contains satellite status as well as the current date and time.
This data updates every two hours and is valid for four hours.
3. Almanac data contains the position of the satellite and is updated every
24 hours.
6.2 Satellite Orbits
GPS SIS
Right-hand circularly polarized
1575.42 MHz & 1227.6 MHz
NAV data
upload
from CS
Atomic
frequency
standard
TT&C
subsystem
Frequency synthesizer
and distribution unit
Navigation
data unit
• 10.23 MHz synthesized
digital clock
• NAV & control
data checks
• 50 Bits/s
NAV data
Helix array
antenna
Navigation
baseband
• P(Y)-code generation
• C/A-code generation
• Modulo-2 addition of
codes and NAV data
L-band
subsystem
• Spread spectrum
modulation of 1575.42
MHz and 1227.6 MHz
L-band carriers
Figure 6.12 GPS signal generation. Source: Courtesy of United States Government [26].
www.usa.gov/government-works.
The GPS satellite phased array of helical antennas transmits RHCP signals at
two primary GPS frequencies at L band [27]:
• L1 at 1575.42 MHz provides the course-acquisition (C/A) and encrypted precision (P) codes. It is also used for the L1 civilian (L1C) and military (M)
codes.
• L2 at 1227.60 MHz carries the P code, as well as the L2C and military codes.
They transmit sufficient power to guarantee a minimum signal power at the
Earth surface of −166 dBW.
The GPS signals use CDMA with a unique high-rate PRN Gold code for each
satellite [28]. A GPS receiver must have the Gold codes from all the satellites in
order to decode the signal. Anyone has access to the C/A code transmitted at
10.23 million chips per second (Mcps). Only the US military has access to the P
(precision) code that has a rate of 10.23 Mcps. Both codes send the exact time
to the receiver. Only the L1 signal carries the C/A code, whereas the P code
appears in both L1 and L2.
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6 Satellite Communications
Significant subframe contents
Subframe 1 TLM HOW
GPS Week Number, SV Accuracy and Health,
and Satellite Clock Correction Terms
Subframe 2 TLM HOW
Ephemeris Parameters
Subframe 3 TLM HOW
Ephemeris Parameters
Subframe 4 TLM HOW
Almanac and Health Data for Satellites 25–32, Special
Messages, Satellite Configuration Flags, and lonospheric
and UTC Data
Pages
1–25
Subframe 5 TLM HOW
Almanac and Health Data for Satellites 1–24 and Almanac
Reference Time and Week Number
Pages
1–25
Frame
Figure 6.13 Navigation message content and format [26]. Source: Courtesy of United States
Government. www.usa.gov/government-works.
GPS signals transmit frames containing 15 000 bits at 50 bps over 30 seconds
[26]. Frame transmission starts on the minute or half minute according to the
atomic clock on each satellite. A frame consists of five 300-bit subframes that
each take six seconds to transmit. Subframes contain ten 30-bit words that are
0.6 seconds long. The first two words in each subframe carry telemetry (TLM)
and handover (HOW) information. Figure 6.13 lists the data contained within
a frame.
6.3 Satellite Link Budget
Space exploration satellites communicate with the Earth ground stations for
control and data transfer. Figure 6.14 has a graph of the 1/R2 loss between a
ground station and satellites at planets in our solar system. Higher frequencies suffer more atmospheric loss, so the transmitter must have more power
than at low frequencies. A high power transmitter weighs more, takes up more
space, and consumes more operating power than a low power transmitter, so
the satellite transmits at a lower frequency than the ground station.
Size and weight constraints on satellites force the ground station to have the
large antenna, the high power transmitter, and the sensitive receiver to make
up for the huge propagation loss. Reflector antennas or phased arrays provide
the high gain needed by the antennas in a satellite link. Parabolic dish antennas
6.3 Satellite Link Budget
300
Attenuation (dB)
250
Neptune
Saturn
Mercury
Mars
Venus
GEO
MEO
LEO
200
150
100
Pluto
Uranus
Jupiter
0
1
2
3
4
5
6
× 1013
Distance (m)
Figure 6.14 Attenuation as a function of distance from the Earth.
are significantly cheaper than phased arrays, so they serve as the workhorses
of satellite communications [29]. Phased arrays play an important role in
situations that demand high performance like multiple beams or adaptive
nulling [30].
Example
NASA launched the Cassini spacecraft in 1997 in order to explore Saturn.
The large 4-m Cassegrain dish antenna at the end of the Cassini spacecraft
(Figure 6.16) sends data to a dish antenna in the NASA Deep Space Network
(DSN) at 8.425 GHz. It has dual high power amplifiers (HPAs) that deliver
up to 40 W of power to the antenna. When Cassini was 1.5 billion km from
the Earth, then what is the power received by the (i) 70 m (Figure 6.15a) and
(ii) 34 m (Figure 6.15b) reflector antennas in the DSN.
(
Solution
4𝜋At
𝜆2
)(
4𝜋Ar
𝜆2
)
𝜆2
Pt Gt Ae
Pt Gt Gr 𝜆
PAA
=
=
= t t 2r
4𝜋R2
(4𝜋R)2
(4𝜋R)2
(𝜆R)
10
3 × 10
The DL has a wavelength of 𝜆 =
= 3.56 cm
8.425 × 109
Cassini antenna area: At = 𝜋22 = 12.6 m2
The area of the receive antenna is Ar = 𝜋r = 3848.5 or 907.9 m2
PAA
40(12.6)(3848.5)
Pr = t t 2 r =
= 6.8 × 10−16 = −151.7 dBW
(𝜆R)
(0.0356 × 1500 × 109 )2
PAA
40(12.6)(907.9)
= 1.6 × 10−16 = −157.9 dBW
Pr = t t 2 r =
(𝜆R)
(0.0356 × 1500 × 109 )2
2
Friis formula: Pr =
Pt
A combination of the large distance to a satellite from the ground and the low
transmit power available on the satellite leads to a very weak signal arriving at
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256
6 Satellite Communications
Figure 6.15 Cassini spacecraft. Source: Courtesy of NASA. www.nasa.gov.
(a) 70 m
(b) 34 m
Figure 6.16 Reflector antennas in the NASA DSN at Goldstone. (a) 70 m and (b) 34 m.
Source: Courtesy of NASA. www.nasa.gov
6.3 Satellite Link Budget
the ground station. The satellite has a fixed signal frequency, transmitter power,
and transmit antenna gain due to its limited size and high cost. The ground
station on the other hand, has more latitude in size and budget. If the satellite
is a distance R from the receiver on the ground, then the SNR at the ground
receiver is the ratio of the signal power from the Friis formula to the thermal
noise power.
Pt Gt Gr 𝜆2
)( )
(
Ps
Pt Gt 𝜆2
Gr
(4𝜋R)2
=
SNR =
=
2
Pn
kTB
kB(4𝜋R)
T
⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ ⏟⏟⏟
Satellite Ground
transmitter receiver
(6.3)
Note that the receiver has no control over the satellite transmitter factor in
(6.3). The second factor, G/T, depends on the antenna and LNA design at the
Earth station. As a result, satellite ground stations determine the antenna G/T.
Increasing G increases the received signal power, while decreasing T decreases
the noise power. Thus, increasing G/T enhances a wireless system’s probability
of detecting the received signal and is an important system specification.
The noise power collected by an antenna comprises [31]:
• Atmospheric attenuation noise caused by absorption and re-radiation of signal energy by water and oxygen molecules in the atmosphere. This noise
rapidly increases with decreasing antenna elevation angle and precipitation,
because the signal has to travel further through the atmosphere.
• Noise radiated by the Earth that enters through the antenna sidelobes.
• Cosmic (galactic) noise.
• Ohmic losses or losses due to the resistance of the feed system and the
antenna reflectors.
• Attenuation between the antenna and the LNA – essential to put the LNA
very close to the antenna.
Figure 6.17 presents some of the contributors to the antenna temperature.
The antenna system temperature, T ant , is the system noise temperature in
the denominator of G/T and consists of the temperature of external sources
(T aext ), temperature of antenna losses (T aloss ), and temperature of antenna feed
line (T afeed ) [32].
Tant = Taext + Taloss + Tafeed
(6.4)
The antenna temperature due to external sources comes from integrating the
external temperature weighted by the antenna gain over a spherical surface [33].
Taext =
1
4𝜋 ∫0
𝜋
2𝜋
∫0
G(𝜃, 𝜙)T0 (𝜃, 𝜙) sin(𝜃)d𝜃d𝜙
(6.5)
257
258
6 Satellite Communications
Star
Cosmic
Clouds
Sun
Clear night
Taext ≈ 5 K
Taext ≈ 275 K
Rain
Antenna
Atmosphere
Tafeed
Receiver
Taloss
Feed
Taext ≈ 270 K
Surface emissions
Antenna temperature
Tant = Taext + Taloss + Tafeed
Figure 6.17 Contributions to antenna noise temperature.
where T 0 is the blackbody equivalent temperature. Temperatures range from
a few degrees Kelvin to over 290∘ K depending upon where the antenna main
beam points. Typically, the antenna loss and feed temperatures are small.
Example
An antenna has a gain of 40 dB and is 60% efficient. Calculate the G/T when the
lossless feed cable is connected to the receiver that has NF = 3 dB for (a) clear
night (T aext = 5 K) and (b) rain (T aext = 275 K).
Solution
(
Taloss =
)
1
− 1 290 = 193.3 K
0.6
Trec = (103∕10 − 1)290 = 288.6 K
(a) G∕T =
1040∕10
= 15.2 ⇒ 13.1 dB
193.3 + 288.6 + 5
(b) G∕T =
1040∕10
= 13.2 ⇒ 11.2 dB
193.3 + 288.6 + 275
6.5 Multiple Beams
6.4 Bent Pipe Architecture
A communications satellite transponder goes by the name of bent pipe
architecture. One ground station sends a signal at a high frequency to the bent
pipe satellite, and the satellite changes the signal to a lower frequency and
retransmits through a HPA to another ground station. The satellite serves to
redirect the signal from one ground station to another. Figure 6.18 diagrams
the operation of a typical bent pipe satellite.
6.5 Multiple Beams
Rather than using one broad antenna beam (low gain) to cover a large area,
a satellite often has multiple high-gain narrow beamwidth beams from one
antenna array to cover the same area. The multi-beam approach handles
higher data rates and lower powered UL transmitters like those on handsets.
Figure 6.19 shows the beam routing for 4 UL and 4 DL beams [34]. Figure 6.20
illustrates a hypothetical example of a satellite having 14 beams covering the
United States.
The Ka-band SPACEWAY satellite system consists of multiple geo-synchronous satellites operating over a 500 MHz Ka-band bandwidth sub-divided into
®
Frequency
converter
HPA
lin
Up
k
lin
wn
Do
k
LNA
HPA
LNA
Modulator
Demodulator
Data
Data
Figure 6.18 Bent pipe architecture for a satellite communications system.
259
260
6 Satellite Communications
Uplink
beams
Beam routing matrix
Downlink
beams
1
1
2
2
3
3
4
4
Figure 6.19 Beam routing matrix for 4 UL and 4 DL beams.
Figure 6.20 US coverage with 14 beams.
6.6 Stabilization
102 103 104
101
91 92 93 94 95 96 97 98 99 100
76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
F1
LHCP 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
F2
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
F1
RHCP
17 18 19 20 21 22 23 24 25 26 27 28 29 30
7
F2
8
9
10 11 12 13 14 15 16
3
4
1
108
109
5
6
2
106
107
CARACAS BOGOTA
HAWAII
110
111
SAO PAULO/ LIMA
RIO
112
BUENOS
AIRES
Figure 6.21 Polarization and frequency assignment for the Spaceway beams. Source:
Reprinted with permission of Whitefield et al. 2006 [35]. © 2018, IEEE.
16, 62.5 MHz subbands with 8 LHCP and 8 RHCP using FDMA and TDMA
[35, 36]. The satellites use 112 UL cells positioned over the US, Puerto Rico,
and several major cities in Central and Latin America as shown in Figure 6.21.
The DL operates from 19.7 to 20.2 GHz and the UL from 29.5 to 30.0 GHz.
Each UL cell has a 0.5∘ fixed beam with the opposite polarization of the DL
beam. Seven DL microcells with 0.189∘ beam widths lie within a UL cell and
have a polarization opposite to that of the UL cell. These microcells improve
approximately 5 dB DL beam gain loss at the cell edge to less than 1 dB. Adjacent cells have either different polarizations (RHCP or LHCP) and/or different
frequencies (F1 or F2) to allow cell reuse.
6.6 Stabilization
Spin and three-axis body stabilization keep the antennas and solar panels pointing in the desired directions. Spin stabilization works for satellites that have
symmetry about one axis like the model of the Canadian Alouette satellite in
Figure 6.22a, while three-axis body stabilization works for any satellite – no
symmetry needed (TDRS in Figure 6.22b).
261
262
6 Satellite Communications
(a)
(b)
Figure 6.22 Satellite stabilization (a) spin and (b) three-axis body.
Spin stabilized satellites fire jet thrusters to start the spinning once in orbit.
The satellite spins between 30 and 120 rpm about its axis of symmetry [17].
Spinning induces a gyroscopic effect that keeps the spin axis pointing in
a desired direction with little wobble. Spin stabilization requires a design
that has antennas and solar panels symmetrically placed about the satellite.
Traditionally, the solar panels wrap around the satellite body. Placing multiple
extended flat panels around the satellite body works as well.
Three-axis stabilization keeps the satellite in a stationary position relative to
its orbit in order to maintain constant antenna and solar panel pointing accuracy. Satellites have three axes that need stabilization: pitch, yaw, and roll. The
yaw axis points toward the Earths center, the pitch axis is normal to the orbital
plane, and the roll axis is tangent to the orbit. Each axis either has a large flywheel or reaction wheel for stabilization. Reaction wheels keep the satellite
much more steady than flywheels and thrusters, but their weight shortens the
satellite’s lifespan [17]. Sensors monitor external references, such as the sun or
stars, then a controller uses the information to adjust the satellite orientation
by controlling the reaction wheel spin or firing the thrusters on the appropriate axis.
Problems
6.1
Using trigonometry, show that a geostationary satellite with an antenna
that has a 17.3∘ beamwidth covers approximately 1/3 of the Earth surface
outside of the polar regions.
6.2
Find the minimum one-way signal latency (time delay) for an Earth station on the equator communicating with a geostationary satellite.
6.3
Calculate the number of days that significant levels of Sun interference
will be experienced at each equinox for an 11-m diameter antenna at
11 GHz.
References
6.4
Approximate the maximum duration of a Sun transit at each equinox for
an 11-m diameter antenna at 11 GHz.
6.5
An Earth station has an antenna temperature of 45 K that feeds to an
LNA with T = 100 K and G = 50 dB, followed by a mixer with T = 1000 K.
Find the system temperature.
6.6
Find the CNR for a geosynchronous satellite operating at 6.1 GHz
having an antenna with a gain of 26 dB. Its receiver has T = 500 K,
B = 36 MHz, and gain of 110 dB. The Earth station transmits 100 W via
a 54 dB antenna gain.
6.7
Calculate the gain of a satellite antenna having orthogonal beamwidths
of 3∘ and 6∘ at the edge of its coverage zone.
6.8
Find the CNR at an Earth station with an antenna gain of 53 dB receiving
a signal at 3.875 GHz from a satellite at an orbit of 39 000 km, transmitting 10 W through an antenna with a gain of 30 dB.
6.9
Find the CNR at a ground station with Gr = 1 dB, T = 260 K, and
B = 20 kHz. The satellite is 2000 km away and transmits a 2.5 GHz signal
at 0.5 W through an antenna with Gt = 18 dB.
6.10
A satellite system operating at 4.15 GHz has an Earth station antenna
30 m in diameter and an aperture efficiency of 68%. The system noise
temperature varies between 79 and 88 K depending on weather conditions. Find the variation in G/T at the Earth station.
6.11
A satellite at a distance of 40 000 km communicates with a ground station
at 4.0 GHz with a 2 W transmitter and a 17-dB antenna gain. Find the
power received by a ground station antenna having an effective area of
10 m2 .
References
1 http://www.inetdaemon.com/tutorials/satellite/communications/frequency-
bands (accessed 26 August 2017).
2 Clarke, A.C. (1945). V2 for ionosphere research? Wireless World L1 (10):
305–308, Letters to the Editor.
3 Smil, V. (2017). Sputnik at 60. IEEE Spectrum: 20.
4 https://www.nasa.gov/multimedia/imagegallery/image_feature_559.html
(accessed 24 August 2017).
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5 Jakes, W.C. (1961). Participation of bell telephone laboratories in project
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7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
echo and experimental results. The Bell System Technical Journal 40 (4):
975–1028.
http://www.arrl.org/news/oscar-i-and-amateur-radio-satellites-celebrating50-years (accessed 24 August 2017).
https://airandspace.si.edu/collection-objects/communications-satellite-telstar
(accessed 24 August 2017).
https://www.nasa.gov/topics/technology/features/telstar.html (accessed 24
August 2017).
Shennum, R.H. and Haury, P.T. (1963). A general description of the Telstar
spacecraft. The Bell System Technical Journal 42 (4): 801–830.
https://appel.nasa.gov/2010/02/25/ao_1-7_sf_history-html (accessed 24
August 2017).
https://nssdc.gsfc.nasa.gov/nmc/spacecraftDisplay.do?id=1963-004A
(accessed 24 August 2017).
https://nssdc.gsfc.nasa.gov/nmc/spacecraftDisplay.do?id=1963-031A
(accessed 24 August 2017).
Williamson, M. (2006). Spacecraft Technology: The Early Years. London: IET.
Yuan, J., Yang, D. and Sun, X. (2006). Single access antenna pointing control
system design of TDRS (pp. 5–1097). 2006 1st International Symposium on
Systems and Control in Aerospace and Astronautics, Harbin.
https://en.wikipedia.org/wiki/Low_Earth_orbit (accessed 24 October 2017).
Clarke, A.C. (1945). V2 for ionosphere research? Wireless World L1 (2): 58,
Letters to the Editor.
H. Hausman, "Fundamentals of Satellite Communications, part 1," https://
www.ieee.li/pdf/viewgraphs/fundamentals_satellite_communication_part-1
.pdf (accessed 25 October 2017).
https://www.itu.int/dms_pubrec/itu-r/rec/s/R-REC-S.1525-1-200209-I!!PDFE.pdf (accessed 3 January 2018).
Haupt, R.L. (2010). Antenna Arrays: A Computational Approach. Hoboken,
NJ: Wiley.
https://www.airspacemag.com/space/how-things-work-space-fence180957776 (accessed 25 February 2019).
Sekiguchi, K. (2016). Iridium contributes to “maritime safety” (pp. 90–92).
2016 Techno-Ocean, Kobe.
Leopold, R.J. and Miller, A. (1993). The IRIDIUM communications system
(pp. 575–578). 1993 IEEE MTT-S International Microwave Symposium
Digest, Atlanta, GA.
http://spaceflight101.com/spacecraft/iridium-next (accessed 31 May 2018).
https://www.gps.gov/systems/gps (accessed 4 February 2019).
https://www.gps.gov/multimedia/images (accessed 31 May 2018).
US Department of Defenses and GPS NAVSTAR (2008). Global positioning
system standard positioning service signal specification, 4e. GPS NAVSTAR.
References
27 https://www.navtechgps.com/gnss_facts (accessed 26 February 2019).
28 Holmes, J.K. and Raghavan, S. (2004). A summary of the new GPS IIR-M
29
30
31
32
33
34
35
36
and IIF modernization signals (Volume 6, pp. 4116–4126). IEEE 60th Vehicular Technology Conference – VTC2004-Fall.
Rahmat-Samii, Y. and Haupt, R.L. (2015). Reflector antenna developments: a
perspective on the past, present and future. IEEE AP-S Mag 57 (2): 85–95.
Haupt, R.L. and Rahmat-Samii, Y. (2015). Antenna array developments: a
perspective on the past, present and future. IEEE AP-S Mag 57 (1): 86–96.
Ho, C., Kantak, A., Slobin, S., and Morabito, D. (2007). Link analysis of
a telecommunication system on earth, in geostationary orbit, and at the
moon: atmospheric attenuation and noise temperature effects. IPN Progress
Report 42-168, Jet Propulsion Laboratory, 15.
Cakaj, S., Kamo, B., Enesi, I., and Shurdi, O. (2011). Antenna noise temperature for low earth orbiting satellite ground stations at L and S band
(pp. 1–6, 17–22). Third International Conference on Advances in Satellite
and Space Communications, Budapest, Hungary (April 2011).
Ulaby, F.T., Moore, R.K., and Fung, A.K. (1981). Microwave Remote Sensing
Active and Passive, vol. 1. Reading, MA: Addison-Wesley Publishing Co.
Elbert, B.R. (2008). Introduction to Satellite Communication. Artech House.
Whitefield, D., Gopal, R. and Arnold, S. (2006). Spaceway now and in the
Future: on-board IP packet switching satellite communication network
(pp. 1–7). MILCOM 2006–2006 IEEE Military Communications Conference, Washington, DC.
Fang, R.J.F. (2011). Broadband IP transmission over SPACEWAY satellite with on-board processing and switching (pp. 1–5). 2011 IEEE Global
Telecommunications Conference – GLOBECOM 2011, Houston, TX.
®
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7
RFID
Automatic identification (auto ID) technologies identify, track, and collect data
about objects of interest. Auto ID includes bar codes, optical character readers,
radio frequency identification (RFID), and some biometric technologies, such
as retinal and finger print scans. They improve data accuracy and reduce the
amount of time and labor needed to manually input data [1]. An RFID system
has a reader that communicates with a tag. The tag either backscatters a signal
(passive system) or transmits a signal (active system) back to the reader where
it is detected. RFID is useful for inventory, tracking packages, identifying lost
dogs, verifying identification cards, measuring temperature, etc. This chapter
introduces RFID system technology.
7.1 Historical Development
The first passive RFID system originated in WWII when German pilots rolled
their planes as they returned to base in order to modulate the backscattered
signal from German radar [2]. This modulated return distinguished them from
Allied aircraft, so that they did not attract German anti-aircraft fire. In 1939,
the British developed the first active identify friend or foe (IFF) system. An IFF
transponder on a British plane emitted a unique modulated signal when illuminated by British radar that identified the friendly plane to the radar operators.
In 1945, the Soviet Union gave a hand-carved replica of the Great Seal of
the United States to the US Ambassador. Inside the carving was a clever passive listening device called “The Thing” or “The Great Seal Bug” [3]. This bug
had a high-Q resonant cavity with a very thin 75 μm membrane on one end
(Figure 7.1). A monopole antenna sticking out of the bottom couples a strong
RF signal into the cavity at the resonant frequency. Anyone talking near the
carving causes the membrane to vibrate and change the size of the cavity which
in turn changes the resonant frequency and modulates the signal retransmitted
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
268
7 RFID
Silver-plated cavity
Threaded back
Tuning post
Coupling disc
Membrane
Threaded cover
Soft insulation
Insulation
Fixating nut
Antenna
Figure 7.1 Great seal bug [3]. Source: Courtesy of NASA. www.nasa.gov.
from the monopole. The Soviet Union demodulated the AM signal, enabling
them to listen to the conversations. This bug had no internal power source and
functioned like a passive RFID tag.
In 1948, Harry Stockman came one step closer to modern RFID by proposing a communication system in which a transmitter sends a carrier signal to
a vibrating reflector. The reflector modulates the backscattered signal that a
receiver detects and demodulates [4]. His ideas for modulating the backscattered signal relied on mechanics similar to “The Thing.” Stockman noted that
modulation with mechanical vibrations limited this approach to low frequencies.
RFID outperforms bar codes for inventory and tracking applications. Joe
Woodland based his barcode idea on Morse code [5]. He and his colleagues
received a patent in 1952, but his idea did not blossom until the laser and
desktop computers were invented [6]. The bar code contains a universal
product code (UPC) that is a 12-digit number unique to a manufacturer (first
six numbers) and type of product (next five numbers) as shown in Figure 7.2.
The data contained within the barcode depends on the relative width of the
black and white bars and not on the physical size of the barcode. Dark bars are
1–3 units wide while white bars are 1–4 units wide. A UPC cannot distinguish
one can of Coke from another can of Coke, but it can distinguish a liter bottle
of Pepsi from a can of Coke. A check digit for error detection at the end of the
barcode comes from the following algorithm [7]:
1. Add the numbers in the odd positions then multiply the sum by 3.
3(0 + 8 + 4 + 0 + 2 + 6) = 60
2. Add the numbers in the even positions.
7 + 7 + 2 + 8 + 9 = 33
7.1 Historical Development
Figure 7.2 UPC number and associated
bar code.
Manufacturer
identification
number
Item
number
Check
digit
3. Add the numbers found in steps 1 and 2 to get the number q. The check digit
plus q equals a number that is a factor of 10.
60 + 33 = 93 so the check digit is 7
If the scanner does not calculate 7 for the check digit, then it demands a rescan.
The first modern far-field RFID patent envisioned a base station corresponding with a transponder that has memory and data processing power [8].
This patent states in the abstract: “In the preferred inventive embodiment,
the transponder generates its own operating power from the transmitted
interrogation signal, such that the transponder apparatus is self-contained.”
Around the same time, Los Alamos National Laboratory developed an RFID
system with a transponder on a truck carrying nuclear materials and readers at
the security gates [9]. The transponder responded to the reader with data that
identified the truck and driver. This system formed the basis for automated toll
payment systems for roads, bridges, and tunnels in the mid-1980s. Around the
same time, Los Alamos also developed a passive UHF RFID tag to track cows.
A 1983 patent outlined an RFID card system [10]: “An automatic identification system wherein a portable identifier, preferably shaped like a credit card,
incorporates an oscillator and encoder so as to generate a programmable pulse
position-modulated signal in the radio frequency range for identification of the
user. The identifier can be made to generate the identification signal constantly
or can be made for stimulated transmission responsive to an interrogation signal. The identification signal can be preset or can be programmable by use of a
programmable memory.” This type of card is now in wide use throughout the
world for public transportation, motel keys, etc.
In the mid-1980s, Fairchild Semiconductor, Motorola, Texas Instruments,
IBM, and DEC developed an RFID monetary system that implanted an RFID
tag in a person’s hand [11]. The tag enabled transactions by connecting to the
269
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7 RFID
appropriate financial institution. Implanting tags into humans killed this idea.
The resulting patents spurred the very successful pet tagging industry for identifying and recovering lost pets.
IBM developed and patented a UHF RFID system that had a long range and
high data rate in the early 1990s [2]. Wal-Mart partnered with IBM to create
a commercial system that never came to fruition. In the end, a barcode company called Intermec bought all of the IBM RFID patents. The Intermec RFID
systems were used in applications from warehouse tracking to farming. Low
volume and high cost limited sales, though. The lack of international standards
discouraged wide-spread use.
In 1999, RFID technology exploded after MIT created the Auto-ID Center
[12]. This center stimulated the wide acceptance of RFID through the introduction of the Electronic Product Code (EPC). This innovation tracked objects
using the Internet. In 2003, EPCglobal replaced the Auto-ID Center in order to
promote the EPC standard. It manages the EPC network and standards, while
its sister organization, Auto-ID Labs, manages and funds research on EPC technology.
7.2 RFID System Overview
RFID systems have a reader and a tag [13]. The reader and tag communicate
through a sequence of commands (called the inventory round) that leads to the
tag identification and sometimes an exchange of data. Readers communicate
with tags in an area called the interrogation zone (IZ) where the reader signal
level exceeds the tag sensitivity. An RFID reader transmits an encoded radio
signal that interrogates the tag. The RFID tag in the IZ responds with a carrier
modulated by data stored in memory. The data might be a serial number, time
stamp, or configuration instructions which aids in taking inventory or directing
a package to be moved to a certain location.
Figure 7.3 shows block diagrams of monostatic and bistatic RFID systems.
A monostatic reader’s transmitter and receiver share an antenna. The circulator in the reader directs the transmitted signal to the antenna and the received
signal to the receiver while isolating the transmit and receive circuits. Circulators protect the sensitive low power receive circuitry from the much higher
powered transmit signal as well as allows the transmitter and receiver to operate at the same time. Bistatic RFID eliminates the circulator by using separate
antennas for transmit and receive. Bistatic systems have high isolation between
the transmit and receive circuits.
An inlay is the tag’s guts: antenna and circuit on a substrate [14]. A smart
label has an inlay inside of a barcode label with adhesive for easy attachment.
The label surface contains information such as sender’s address, destination
address, and product information. Smart labels are cheap because they are
7.2 RFID System Overview
Monostatic
RFID tag reader/interrogator
Antenna
Transmitter
Controller
LO
Circulator
Receiver
Antenna
RF
channel
Bistatic
RFID tag reader/interrogator
RF
ID
tag
Transmitter
Antenna
Controller
LO
Receiver
Figure 7.3 RFID system diagram for monostatic and bistatic RFID readers.
printed in large quantities. Hard tags made from polycarbonate, ceramic, steel,
polystyrene, and polypropylene are rigid and thicker than labels. They cost
much more than smart labels and are attached to an object by adhesive, shrink
wrap, stitching, straps, screws, or other means.
RFID tags fall into two broad categories: near field and far field (Table 7.1).
Near-field tags operate close to the reader at low frequencies and low data
rates. They use loop antennas to couple LF or HF magnetic field between the
reader and tag. Low frequencies penetrate materials well, so the environment
has little impact on the signal. In contrast, far-field tags use dipole antennas
and communicate at much higher frequencies and data rates. They communicate with the reader over much larger distances. Their fields have small penetration depths into materials, so the environment significantly impacts the
signal.
Tags are either active, passive, or semi-passive. Active tags are transponders
with their own power source that boosts the signal power sent to the reader
and increases the communication range. Passive tags harvest energy from the
reader’s signal in order to power the electronics that retrieves data from memory, then modulates the signal scattered back (backscatter) to the reader. The
cheap passive tags have limited range. Semi-passive or battery-assisted tags
(BATs) use battery power for data collection and processing but not for boosting the signal strength sent to the reader. The best type of tag to use in a system
depends on the physical environment, required read range, and material properties of the tagged object.
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7 RFID
Far field
Near field
Table 7.1 RFID frequency allocations [15–17].
Frequency band
Range
Tag type
Data rate
Applications
120–150 kHz (LF)
10 cm
Passive
Low
13.56 MHz (HF)
10 cm–1 m Passive
Low to
moderate
• Access control
• Livestock
tracking
• Ticketing
• Payment
• Data transfer
• Airline baggage
• Libraries
433 MHz (VHF)
1–300 m
Active
High
Europe: 865–868 MHz
North America:
902–928 MHz (UHF)
1–12 m
Moderate to
Passive
Semi-passive high
Active
2450–5800 MHz
(microwave)
1–40 m
Passive
High
Semi-passive
Active
•
•
•
•
•
Tracking
Locating
Parking lot access
Toll collection
Supply chain
• Toll roads
• Vehicle
identification
• Supply chain
RFID systems follow international and regional standards. The ISO (International Organization for Standardization) has a series of standards defining the
interface between the reader and tag [18]:
1. 18000-1: Generic Parameters for the Air Interface for Globally Accepted Frequencies
2. 18000-2: Parameters for Air Interface Communications below 135 kHz
3. 18000-3: Parameters for Air Interface Communications at 13.56 MHz
4. 18000-4: Parameters for Air Interface Communications at 2.45 GHz
5. 18000-5: Parameters for Air Interface Communications at 5.8 GHz (Withdrawn)
6. 18000-6: Parameters for Air Interface Communications at 860–960 MHz
7. 18000-7: Parameters for Air Interface Communications at 433 MHz
Governments throughout the world already assigned UHF frequencies around
900 MHz long before RFID, so no internationally recognized frequency band
exists for RFID. As a result, the newest Gen 2 (second generation) protocol
works at a narrow frequency band between 860 and 960 MHz.
Communication between reader and tag is either full duplex (FDX) or half
duplex (HDX) [19]. In an FDX system, the reader and tag transmit and receive
data at the same time by using different frequency bands. In an HDX system,
either the reader or tag transmits data, but not both at the same time. An HDX
reader charges a capacitor in the tag in the initial wakeup. After the reader stops
7.3 Tag Data
transmitting, the tag uses power from the charged capacitor to transmit the
requested data to the reader. An HDX reader uses simpler decoding techniques
than FDX. A kill command from the reader permanently stops all tag functions.
An inoperable tag is called a quiet tag [14].
7.3 Tag Data
A UPC code enables a grocery store to count bottles of Pepsi sold but cannot
help Amazon track the purchase and delivery of a customer’s item. Tracking
individual items requires a much more sophisticated code. The EPC solves this
problem by incorporating information about an individual item with the UPC.
The EPC is a unique universal identifier assigned to every product and all
categories of products in the world [20]. It contains the information indicated
in Figure 7.4 [21]. The header identifies the information structure encoded on
the tag, including the type of EPC and the encoding length (64, 96, or 128 bits).
Additional instructions for the reader follow the header. The filter value tells
the Reader to select or disregard certain tags, while the partition value tells the
reader where the company prefix ends and the item reference begins. The EPC
manager number identifies the entity that maintains the remaining partitions
of the EPC. An EPC Manager assigns a variable-length object class and a serial
number to a specific instance of that product. Figure 7.5 is an example of the
last three parts of the EPC binary encoding shown in Figure 7.4.
Added instructions
for reader
Header
Filter
value
Partition
value
Defined in tag
data standard
EPC
manager
number
Assigned by
EPC global
Object
class
Serial
number
Assigned by
EPC manager
owner
Figure 7.4 EPC binary encoding on RFID tag.
Figure 7.5 EPC code example.
345+
84116
Header Company
code
15521
Product
code
+1234567890 = EPC
Serial
number
EPC = 34584116155211234567890
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7 RFID
Some tags use electrically erasable, programmable, read-only memory (EEPROM) that does not require continuous power to store data [22]. A tag with
EEPROM stores the data for a long period (several years), even without any
power. The type of data stored in the tag falls into one of the four categories:
1. EPC: Electronic Product Code.
2. User: user defined data about the item, such as item type, last service date,
or serial number (32 to over 64 000 bits).
3. Reserved: access and lock passwords that limit viewing and editing data.
4. TID, tag identifier: unique random number provided by the manufacturer
and cannot be changed. The reader requires special settings in order to read
this number (instead of the EPC).
Tag read-only (RO) memory resembles a bar code, because it contains a small
amount of static data that cannot change. A write-once-read-many (WORM)
tag is programmed only once. A RO or WORM tag contains information like
manufacturing date or location. Data in read–write (RW) or smart tags can be
modified many times. A pallet may have both RO and RW memory in which
the RO memory contains the pallet serial number and the RW indicates the
contents at a particular time.
7.4 Tag Classes
A tag needs power for its electronics and signal generation and transmission.
Passive tags derive their power from the reader signal and fall under Class 0–2.
The battery in a semi-passive tag (Class 3) powers electronics but does not help
amplify the signal returned to the reader. Active tags get their power from a
battery and fall under Class 4 or 5.
7.4.1
Passive Tags
A passive tag uses energy harvesting to extract operating power from the reader
signal. This small amount of power runs the integrated circuit (IC) and powers
the backscattered signal. A passive tag outside the IZ cannot harvest sufficient
power to properly function. The lack of a continuous power source limits the
amount of the data transmitted by a passive tag. Passive RFID tags fall into three
of the six standard classifications [23]:
• Class 0: A passive tag with RO memory used to detect the tag’s presence.
A 0+ tag is a write once/read-only memory (WORM) tag.
• Class 1: A passive, WORM, backscatter tag with a one-time, field-programmable, nonvolatile memory. Tag data supplied by the manufacturer or user.
• Class 2: A passive backscatter tag that has identification as well as other information in memory.
7.4 Tag Classes
Passive tags are ideal for nonreusable applications. Some advantages of passive
tags include: [24]
•
•
•
•
•
•
Small size
Lightweight
Inexpensive (depends on quantity)
Generates no RF noise
Longer life (20-plus years)
Resists harsh environment
Disadvantages of passive tags include:
•
•
•
•
Requires a reader to supply the power
Has limited data storage
Sends a weak signal to reader
Has a short read range (several cm to several m)
RF-DC power conversion efficiency (𝜂 PCE ) equals the power arriving at the
load divided by the power received by the antenna. The rectifier, voltage multiplier, and storage elements contribute to 𝜂 PCE . For a passive tag, 𝜂 PCE Pr must
exceed the tag sensitivity in order to enable a response. The energy harvesting
circuit charging time depends on distance from reader, antenna size, inductance, Q factor, and charging station field strength.
Passive tags balance the power needs of the IC in the tag with the backscattered power needed to exceed the sensitivity of the reader. The tag antenna
passes the reader signal to a matching circuit then through a circulator to the
demodulator and energy harvesting circuit (Figure 7.6). Next, the DC output
from the harvesting circuit provides power to the memory, modulator, and
demodulator. A harvesting circuit or charge pump converts the AC signal into
DC then amplifies it using a voltage multiplier. Schottky diodes with low series
resistance or metal–oxide–semiconductor field effect transistors (MOSFETs)
produce a high 𝜂 PCE [25]. A large energy-storage capacitor bank smooths the
rectifier output waveform and converts it into a nearly DC signal. In addition,
Baseband data
RF
Modulator
Memory/controller
Antenna
Matching
circuit
Demodulator
Circulator
Baseband data
Power
harvesting
Figure 7.6 Passive RFID tag.
DC
275
276
7 RFID
it stores enough energy to operate the tag when the reader signal is low or
not present. The rectifier’s power conversion efficiency is defined for a voltage,
V load , across a load resistance, Rload [26]:
𝜂PCE =
2
Vload
(7.1)
Rload Pin
where Pin is the received rectifier power. This power enables the demodulator
to pass a baseband signal to the memory/controller and the memory/controller
to send its data to the modulator that passes an RF signal to the antenna for
transmission back to the reader.
The DC voltage (V DC ) of the Dickson multiplier or Dickson charge pump in
Figure 7.7 depends on the number of stages [27]. Adding more stages increases
the output voltage but decreases the efficiency [28]
𝜂PCE =
VDC
VDC + 2NV D
(7.2)
where N = number of stages and V D = forward voltage drop. For a Schottky
diode, 0.15 ≤ V D ≤ 0.45 V. The Dickson multiplier has the disadvantages of
requiring multiple stages for a usable supply voltage and may also require large
passive components (e.g. large capacitors) to reduce the output ripple.
Other circuits similar to the Dickson multiplier are referred to as voltage
doublers or multipliers. Another approach uses bridge rectification circuits.
Major considerations in choosing the right rectifier circuit include impedance
matching between the rectifier and antenna, the input power level to the
+
Antenna
Capacitor
+
Stage 1
Diode
Transient
Stage 2
Steady state
VDC
VAC
Stage N
–
–
t
Ground
Figure 7.7 Diagram of a Dickson charge pump and its output voltage.
7.4 Tag Classes
rectifier, the DC power and DC voltage required, and the RF frequency. An RF
energy-harvesting architecture has the advantage of fitting on the IC [15].
The Federal Communications Commission (FCC) limits transmit power in
the band 902–928 MHz to less than 1 W. Consequently, link analysis concludes
that a reader must deliver between 10 and 30 μW to the tag IC in order to activate the tag. If the charge pump is about 30% efficient, then tags with perfectly
matched antennas need their antenna to deliver 30–100 μW in order to power
the chip.
Example
Assume the 915 MHz tag IC has a threshold power level of −10 dBm. The reader
transmits 30 dBm through an antenna with 6 dB gain. Assume the tag antenna
is isotropic and 𝜂 PCE = 0.3. What is the maximum extent of the IZ?
Solution
Start with the Friis transmission formula then solve for r:
P G c2
Pr = t t 2 𝜂PCE
(4𝜋rf )
√
√
Pt Gt 𝜂PCE
c
3 × 108
0.3 × 1 × 4
r=
=
= 2.86 m
4𝜋f
Pr
0.0001
4𝜋(915 × 106 )
7.4.2
Tags with Batteries or Supercapacitors
Either a battery or supercapacitor provides power to active and semi-active tags
[26]. A battery has a high energy storage capacity, while supercapacitors have a
large capacitance that stores a significant amount of electrical charge. Usually,
a supercapacitor has less energy-storage capacity than a battery, but smaller
parasitics makes it attractive. Larger tags with big IZs need large batteries that
require replacement or recharging. A typical active or semi-passive tag uses a
lithium-thionyl chloride battery [29]. These cells offer high energy density and
a long shelf life, and withstand extreme temperatures better than lithium-ion
batteries. They are not rechargeable, though.
7.4.2.1
Semi-Passive Tags
A Class 3 tag is a semi-passive or BAT that uses a battery instead of an energy
harvesting circuit to power the IC [23]. A semi-passive tag has a longer read
range than a passive tag, because the backscattered signal uses 100% of the
harvested power, since the battery only powers the IC. Tags with sensors need
batteries to continuously operate in the absence of a reader signal. A tag sensor
might measure temperature, pressure, relative humidity, acceleration, vibration, motion, altitude, chemicals, or other physical attributes.
277
278
7 RFID
Reusable plastic containers in manufacturing and food-processing justify the
expense of semi-passive tags. Some advantages of semi-passive tags include
[24]
•
•
•
•
•
Increased read range
Reduced reader power
Increased memory storage
Supports environmental sensors
Reduced radio noise
while the disadvantages are
•
•
•
•
Sensitivity to harsh environment
Limited battery life
Costs more than passive tags
Larger size and weight than passive tags
The link budget calculated from the Friis transmission formula determines
the amount of battery power needed to operate an active tag. The tag power
received at the reader is given by
Pr =
Pt Gr Gt c2
4𝜋fc2 r2
L
(7.3)
The generic loss constant, L, includes circuit and channel losses. A tag battery
generates Pt that radiates from an antenna with gain, Gt and is received by a
reader antenna with gain, Gr . The maximum range between the reader and tag
is found by solving (7.3) for r when the receiver sensitivity is given by Pr .
7.4.2.2
Active Tags
Active RFID systems primarily operate at 433 MHz (VHF) but also inhabit the
UHF and microwave bands as well. There are two classes of active tags [23]:
• Class 4: Active tags that have a battery to power onboard circuits and sensors
as well as the transmitter.
• Class 5: Same as class 4 tag but also communicates with other classes of tags
and devices. It is also called a reader tag.
Active tags have a range of up to 100 m, and their higher power penetrates
materials that have low conductivity. Their lifespan depends on the battery life.
In general, large objects, such as rail cars, big reusable containers, and other
assets that need to be tracked over long distances need active tags. One application of an active tag monitors temperature within a refrigerated truck [24].
Another common application monitors security and integrity of a shipping
container.
Active tags are either transponders or beacons [30]. Transponders wake up
when they receive a signal from a reader and then respond with their own signal.
7.5 Data Encoding and Modulation
They have a long battery life, since they power off most of the time. Beacons
have higher power requirements, on the other hand, because they continuously
emit a location signal at predetermined intervals in order for the receiver to
precisely locate a tag. In a real time locating system, a beacon emits a signal
with its unique identifier at preset intervals.
7.5 Data Encoding and Modulation
The reader transmits a signal that wakes up and powers a passive tag as well
as sends data. Most small, low cost tags use a simple modulation scheme that
minimizes electronics on the tag, such as ASK.
A reader using OOK encoding will generate a long sequence of zeros at some
point in the data stream. A string of zeros means no RF signal exists, so a passive tag turns off, because it cannot harvest any power. To overcome the OOK
problem, readers encode binary data using pulse-interval encoding (PIE). PIE
is reminiscent of Morse Code, because it uses a long pulse (dash) to represent
a “1” and a short pulse to represent a “0.” A Type A Reference Interval (TARI)
measures the width of the data “0” symbol. TARI values range from 6.25 to 25 μs
(25 μs gives a data rate of 160 kbps) [31]. PIE dedicates more power to each bit
than OOK [29]. Since the one and zero symbols in PIE have a carrier present
as shown in Figure 7.8, the tag harvests power from both zeros and ones.
PIE is less efficient than OOK, because a PIE TARI is 1/3 as long as an OOK
pulse for the same data rate [31]. As a result, the PIE bandwidth is three times
wider than the OOK bandwidth. In order to fit the PIE bandwidth inside a
500 kHz channel, the data rate cannot exceed 85 kbps, which corresponds to
reader data rates in the United States when using unfiltered PIE [31]. OOK has
equal low and high pulses, so it contains 50% of the maximum power delivered to the tag. If PIE has a high pulse that is three times as long as the low
pulse then it delivers about 63% of peak power. The data rate depends on the
data: a message with many zeros transmits faster than a message with many
ones. Increasing the power delivered to the tag comes at the price of a lower
data rate.
m(t) = PIE encoding
Figure 7.8 PIE encoding
with ASK modulation.
t
Data 1
0
1
0
Carrier
279
280
7 RFID
Figure 7.9 Example of FM0 encoding.
t
0
1
1
1
0
Tags encode data using either FM0 or Miller encoding. ASK, FSK, or PSK
then modulates the encoded data.
FM0 encoding has the following rules:
1. Phase inversion at the beginning of each new symbol.
2. “1” is constant over the bit period.
3. “0” has one phase change during the bit period.
Figure 7.9 shows an example of FM0 encoding applied to a binary data stream.
Its data rate ranges from 40 to 640 kbps. For RFID using FM0, an SNR of around
10 dB or more is usually sufficient [32].
Miller coding starts with Manchester coding of the signal in which a NRZ
data bit stream is XORed with a clock at a power of two times the bit rate
[26]. This signal is not spectrally efficient for transmission, though, because it
has many transitions. Miller encoding increases the spectral efficiency by using
these coding rules [26]:
1.
2.
3.
4.
“1” has a phase transition during the symbol period
“0” is constant over the symbol period
No phase transition between symbols unless consecutive zeros
Subcarrier modulation: The baseband Miller encoded waveform is multiplied by a square wave in order to move the information frequency further
from the carrier frequency. The advantage is better interference rejection.
The disadvantage is a reduced data rate.
5. M square wave transitions per symbol: M = 2, 4, 8
Miller coding hardware implementation starts with an XOR gate that generates
a Manchester code followed by a falling-edge-triggered flip-flop divider that
eliminates every other transition. The output drives the backscatter transistor
to modulate the carrier.
Example
Given the data stream: 0 1 1 0 0 1 0 1, plot the baseband signal voltage for NRZ,
Manchester, and Miller coding.
Solution
Figure 7.10 shows the resulting plots. Note that the Miller coding has much
fewer transitions than the Manchester coding.
7.6 Reader-Tag Communication
Figure 7.10 NRZ data in
terms of Manchester and
Miller coding. Source:
Reprinted with permission
of Shan et al. 2016 [26].
© 2018, IEEE.
NRZ
0
1
1
0
0
1
0
1
Manchester 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0
Miller 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 1
7.6 Reader-Tag Communication
Reader-tag communication either takes place in the near field or far field.
Near-field communications (NFCs) uses a loop antenna operating at LF or HF.
Typically, the reader and tag are less than 10 cm apart for LF and 1 m apart for
HF. LF or HF inductive coupling between loop antennas results in slow data
rates. Far-field communication uses dipoles operating at UHF and above to
transfer data through capacitive coupling in the far field.
7.6.1
Near Field
Typical components of an NFC system appear in Figure 7.11. At close distances,
the magnetic field dominates and decreases as 1/r3 in the near field. Figure 7.12
is a plot of the total power radiated by a loop antenna along with its near and
far field components. The demarcation between the near field and far field is
approximately 1/k = 𝜆/2𝜋. An NFC reader radiates a magnetic field that does
not have nulls in its IZ. Power transfer between two loops in NFC depends
on [33]:
operating frequency
number of turns in the coils
area of the coils
angle between the coils: maximum coupling occurs when the loops are in the
same plane
• distance between the coils
•
•
•
•
Advantages of NFC include:
• the short read range protects sensitive data from intrusion. For instance,
credit card tags need a short read range to deny access to other readers.
• the environment has little impact on the antenna performance.
• LF and HF signals penetrate most materials.
• very reliable
281
Using induction for power coupling from reader to tag and
load modulation to transfer data from tag to reader
Magnetic field
affected by tag data
Power and data
(if tag supports
write)
Data via
changes
in field
strength
RFID
reader
RFID
tag
Binary tag ID
Glass or plastic
encapsulation
Coil
c/2πf
Near-field region
Alternating magnetic field in
the near-field region
Figure 7.11 Near-field RFID system. Source: Reprinted with permission of Want 2006 [2]. © 2006, IEEE.
Far-field region
Propagating electromagnetic
waves
7.6 Reader-Tag Communication
0
Relative power (dB)
Figure 7.12 Power decreases as a
function of distance from the loop
antenna.
Total power
Far field power
Near field power
–50
–100
–150
10–2
λ
2π
10–1
100
Distance from loop (λ)
101
While disadvantages are:
•
•
•
•
the short read range requires tags and readers be in close proximity.
a large antenna
low data rate
the inability to discriminate between multiple tags.
LF tags are expensive and thicker than tags at higher frequencies. Figure 7.13
displays two types of LF tags. The multi-turn loop antennas connect to an IC or
to a capacitor for power storage. The LF tag in Figure 7.13a is a flat multi-turn
loop antenna and has a large area. The one in Figure 7.13b is thicker and has an
iron core to minimize the number of turns in the loop [36].
An LF reader takes between 25 and 50 ms to charge a tag [34]. When charging finishes, the reader encodes the data bits using FM0 which modulates a
134.2 kHz carrier for transmission to the tag. The tag responds by switching
its load impedance between open and matched in order to modulate the magnetic field with its stored data. LF tags often use FSK with a 129.3 μs one bit
at 123.7 kHz and a 128.3 μs zero bit at 134.7 kHz. A 12.2 ms 96-bit FSK modulated end of burst (EOB) signal from the tag lets the reader know that the
IC
Multi-turn
loop antenna
Ferrite rod
Antenna
(a)
Figure 7.13 Examples of LF RFID tags.
Capacitor
IC
PCB
Glass housing
(b)
283
284
7 RFID
data was received. A typical tag uses Manchester encoding of a 32-bit unique
ID followed by a 64-bit data stream (Header + ID + Data + Parity). If the tag
receives an invalid command, it sends back 16 bits to tell the reader to retransmit the data. All the stored power dissipates by the end of the tag transmission
until the reader transmits another signal.
Example
If f c = 125 kHz and the tag bit rate is f c /32, then find the data rate and the time
needed to receive 64 bits.
Solution
Data rate = 125 kHz/32 = 3.9062 kbps. Receiving 64 bits: 8 μs × 32 × 64 =
16.384 ms.
LF tags function well near water, animal tissues, metal, wood, and liquids.
The automotive industry (largest LF tag user) embeds an LF tag inside the ignition circuit of an automobile vehicle immobilizer system. Placing the key in the
ignition causes an RFID reader to check the tag ID. The car only starts when
the ID is verified.
Passive HF tags have a low data rate and a read range less than 1 m. HF RFID
systems come in two forms: proximity and vicinity [35].
Proximity tags have more data storage and functionality (e.g. encryption, some processing power, and data storage and retrieval) than LF tags
but require more operating power [36]. The power requirements for tag
activation and operation generally limit the IZ to less than 20 cm. An HF
proximity reader operates with 13.56 MHz modulated by a subcarrier at
f c /128 = 105.9375 kHz, f c /64 = 211.875 kHz, f c /32 = 423.75 kHz, or f c /16 =
847.5 kHz. The tag modulates the magnetic field with a subcarrier at
f c ± f c /16 = 13.56 ± 0.8475 MHz = 14.4075 and 12.7125 MHz. These subcarriers are modulated at the reader data rate.
Vicinity tags have less power and lower data rates than proximity tags [36].
Vicinity cards also operate at f c = 13.56 MHz but have a maximum IZ that
extends about 1 m. The subcarrier frequency is at 423.75 kHz with f c /32. The
subcarrier is then modulated with FSK or OOK modulation. The data rate is
26.48 kbps. Other data rates include f c /8, f c /16, f c /32, f c /40, f c /50, f c /64, f c /80,
f c /100, and f c /128. They are used in inventory control and theft deterrence
systems.
Unlike LF tags, an HF tag may have anti-collision capability that allows
multiple tags to simultaneously coexist in the IZ without interfering with each
other. HF tags cost less than LF tags due to a smaller antenna design. They come
7.6 Reader-Tag Communication
IC
Antenna
Figure 7.14 Examples of HF RFID tags.
in different sizes, some less than a centimeter in diameter. Water, biological
tissues, metal, wood, and liquids have little impact on HF tag performance, but
metal in close proximity to the tag detunes the antenna. HF readers are less
complex and expensive than readers at higher frequencies. An HF smart card is
a plastic RFID tag about the size of a credit card (54 mm× 85.5 mm× 0.8 mm).
A typical HF card uses Miller encoding with subcarrier modulation at 847 kHz
(f c /16). HF RFID systems are the most widely used worldwide. Figure 7.14
shows two examples of HF RFID tags. The tag’s memory has data to control
access, collect transportation fees, store medical data, make purchases etc. It
can be refreshed in order to add money, change access, or modify data.
7.6.2
Far Field
Far-field systems use dipoles instead of loops to transmit and receive signals
at much higher frequencies and data rates and over larger distances compared
to the NFC systems (Figure 7.15). Far-field tags modulate the backscattered
field by changing the antenna impedance. Sending the stored data to the transistor gate switches the load connected to the dipole and creates a modulated
backscattered signal. Any impedance mismatch generates a backscattered field
from the reader signal. The transistor does not need to completely short the
antenna to allow some received energy to continue to power the tag. Figure 7.16
is an example of a far-field tag for paying tolls on highways in Colorado, USA.
7.6.2.1
Multiple Readers in an Interrogation Zone
Multiple readers interfere with each other when their IZs overlap. Since
near-field signals decay much faster than far-field signals (1/r3 ≪ 1/r2 ), NFC
readers have very small IZs and generally do not interfere with each other.
285
Using electromagnetic (EM) wave capture to transfer power from reader to tag
and EM backscatter to transfer data from tag to reader
Data modulated
on signal reflected
by tag
RFID tag
Power
RFID
reader
Binary tag ID
Data (if tag supports data write)
Antenna dipole
Near-field region
Propagating electromagnetic waves
(typically UHF)
Far-field region
Figure 7.15 Far-field RFID system. Source: Reprinted with permission of Want 2006 [2]. © 2006, IEEE.
Glass or plastic
encapsulation
7.6 Reader-Tag Communication
Antenna
IC
Figure 7.16 UHF passive RFID tag for paying highway tolls.
UHF and microwave readers, on the other hand, have large IZs that often
require interference countermeasures such as [37]:
•
•
•
•
•
•
Physical isolation
RF absorber
Shielding
Reduced transmit power
Frequency hopping
Band separation
The type of countermeasure depends on the number and closeness of readers
in the environment.
Reader environments fall into one of the three categories:
1. A single reader broadcasting in an available channel is susceptible to interference from nonreader sources. Shielding and making appropriate changes
to the signal strength and antenna gain mitigate the interference.
2. A multiple reader environment has more channels than readers. Multiplexing reduces interference between signals.
3. A dense reader environment has more readers than channels, so interference abounds. Readers and tags have separate channels to prevent strong
reader signals from overpowering the weaker tag signals. Dense reader mode
occurs when more than 50 (North America) and 10 (Europe) readers have
overlapping IZs. No two readers should transmit at the same time and on
the same frequency when their IZs overlap.
Two common approaches to operating in a dense reader mode are frequency
hopping (Chapter 5) and listen before talk (LBT). A UHF reader in North America supports frequency hopping over 50, 500 kHz channels between 902 and
928 MHz (915 MHz band) and resides less than 0.04 seconds in any one channel. Two readers are unlikely to operate at the same frequency within this wide
bandwidth. Frequency hopping does not work for the narrower UHF bands in
Europe and Japan that cannot handle as many hops [38]. In LBT, a reader checks
if the transmitting channel is free. If the channel is busy, the reader tries another
channel. Other approaches to dense reader mode include
• Allocate tags and readers to different channels.
• Assign a time slot to each reader.
• Dynamically assign time slots.
287
288
7 RFID
Reader signal
Tag signal
50 ms 20 ms 20 ms
50 ms
90 ms
20 ms
70 ms
140 ms
Figure 7.17 Synchronization of reader and tag signals.
Singulation means a reader picks out a tag with a specific serial number from
a group of tags in its IZ [39]. The reader needs an anti-collision protocol (e.g.
multiplexing or aloha) that prevents devices from interfering with each other.
Tree walking, the most common approach to singulation, asks all tags with
a serial number that starts with either a 1 or 0 to respond. If multiple tags
respond, then the reader asks for all tags with a serial number that starts with
01 to respond. This process continues until the reader finds the desired tag (the
only one still responding).
Multiple readers that simultaneously operate in the same IZ avoid mutual
interference by synchronizing their transmit and receive signals as shown in
Figure 7.17 [40]. In LF wireless synchronization, the reader transmits for 50 ms
then the tag sends data after detecting an end of the charge burst. The reader
listens for 20 ms before sending the next 90 ms signal. If the reader detects a
signal from another reader it waits (backs off ) 70 ms in order for the read cycle
to finish. The initial reader always starts its next cycle after this 70 ms delay, so
that it does not constantly back off and never reads any tags. The worst case
synchronization time is 70 ms, while the worst case cycle time is 140 ms.
A reader starts communication with a passive tag by transmitting a continuous wave (CW) signal that powers the tag. Once the tag IC has enough
power, it either transfers data to the reader (TTF protocol – Tag Talk First) or
answers a reader query (RTF protocol – Reader Talk First) The choice of protocol depends on the number of tags within the reader’s range. The two protocols
are incompatible, because a TTF tag in an RTF reader IZ disrupts RTF tag communication. The TTF protocol quickly identifies a solo tag in the IZ. The reader
only transmits a CW signal when not communicating with tags, so it reduces
interference with other wireless systems.
7.6.2.2
Backscatter Communication
Figure 7.18 has a simplified diagram of a passive far-field RFID tag and a monostatic reader [41]. The reader generates a carrier signal 1 that passes through the
circulator and bandpass filter (BPF) 2 to the transmitting antenna 3. From there
7.6 Reader-Tag Communication
3
2
fc
4
Zant
Tag
Modulator
BPF
1
5
0,1
ZIC
90°
Γ0,1
0
ZL
Reader
1
ZL
Z0
6
7
VI(t)
LPF
DC
block
ADC
I Output
LPF
DC
block
ADC
Q Output
8
VQ(t)
Figure 7.18 Model for the monostatic reader-tag RFID system.
the signal goes through the channel and encounters multipath, attenuation,
reflection, and diffraction. It arrives at the tag antenna 4 then travels through
the transmission line to the modulating circuit 5. The tag modulator uses the
stored data to toggle the switch between two impedances: ZL0 that represents
a binary zero and ZL1 that represents a binary one. The reflection coefficient
between the antenna and IC is
Γ0,1 =
∗
ZL0,1 − Zant
ZL0,1 + zant
(7.4)
Selecting ZL0 = ∞ (open) results in Γ0 = 1, while ZL0 = 0 (short) results in
Γ0 = − 1. Load switching modulates the backscattered field with the required
data. This modulated signal reflects back to the antenna 4. The antenna
transmits the backscattered signal to the reader 3. The reader receives the
signal through the receive antenna (different than transmit antenna for a
bistatic system). Next, the signal passes through the BPF before the circulator
directs it toward the receiving circuitry 2. Half the signal goes to the I channel
where it is demodulated with the carrier 6. The other half goes to the Q channel
where it is demodulated with the carrier shifted by 90∘ 6. Both signals then
pass through a LPF, DC block, amplifier, and ADC 7 to get the I and Q output
8. More details are given for some of these steps in the following paragraphs.
The reader transmits a carrier signal that activates tags in the IZ before sending a query requesting tags to respond with their identification. In order to read
a passive tag there must be sufficient power transferred to the tag in order to
power the IC. As with any wireless system, calculating the link budget requires
289
290
7 RFID
knowledge of the channel effects. The power delivered to the tag IC is given
by [41]
Gtag 𝛿p Tload
𝜆2
Ptag = Preader Greader
2
⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ (4𝜋r) 𝜂ob Lblock Ftag
⏟⏟⏟ ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟
Reader
Space loss
(7.5)
Tag
where
Preader = reader transmit power
Greader = gain of reader antenna in direction of tag
Gtag = gain of tag antenna in direction of reader
𝛿p
= polarization loss factor
T load = power transmission coefficient
𝜂 ob
= on object gain penalty
Lblock = blockage loss
F tag = fade margin
The tag size, orientation, angle, and placement impact the read range. Steep
angles of incidence reduce the read range due to lower antenna gain and polarization mismatch. Maximum gain and polarization efficiency occur when the
tag and reader antennas face each other. If the reader antenna and tag antennas
are linearly polarized, then 𝛿 p ranges from 0 to 1 depending on the orientations
of the antennas. In order to avoid fades due to antenna orientation, a circularly
polarized reader and a linear polarized tag antenna insures that 𝛿 p = 0.5.
The power delivered to the IC depends upon the match between the antenna
and the IC. Some of the power reflects while the rest goes to the IC based on
the transmission coefficient (0 ≤ T load ≤ 1) [41]:
Tload =
4Re{Zant }Re{ZL0,1 }
Re{Zant + ZL0,1 }2 + Im{Zant + ZL0,1 }2
(7.6)
The input impedance of a tag antenna in free space differs from the tag
attached to an object. This impedance difference increases as frequency
increases. Low dielectric constant objects have much less impact on the tag
antenna input impedance compared to water or metal. Metal-mount tags have
a built-in, low dielectric between the tag and the metal object [40].
Figure 7.19 plots Ptag vs. separation distance for tags in free space, mounted
on cardboard, and mounted on aluminum at 915 MHz [41]. This plot emphasizes that the Friis transmission formula in free space (𝛿 p = 1, T load = 1, 𝜂 ob = 1,
Lblock = 1, F tag = 1) does not adequately calculate the link budget. Mounting the
tag on aluminum causes over 30 dB of additional loss compared to the same
tag mounted on cardboard. Setting the tag threshold at Ptag = −12 dBm (dotted
horizontal line in Figure 7.19) results in an IZ of 2 m for the tag mounted on
cardboard, while the tag on aluminum has an IZ much less than 1 m.
7.6 Reader-Tag Communication
Friis tra
nsmiss
io
n formu
la
Ptag (dBm)
–10
–20
Tag threshold
On card
board w
ith
–30
–40
On alu
minum
with
–50
1
2
multipa
th
multipa
th
4
6
8
10
r (m)
Figure 7.19 The power uplink budget for tags mounted on different materials as a function
of separation distance.
100
Tag
Reader
y (cm)
50
LOS
0
A
B
C
Floor
–50
–100
–200
–150
–100
–50
0
x (cm)
50
100
150
200
Figure 7.20 Reader-tag communications link with two boxes (A and B) blocking the LOS
signal to box C.
Figure 7.20 models a situation where the RFID tag on box C lies behind
boxes A and B. The tag and reader are at the foci of the Fresnel ellipsoids
(Chapter 5). Box A is in zone 0 and blocks the LOS. Box B also blocks the LOS
but extends up into zone 2. As a rule of thumb, when the Fresnel zone number
is small (on the order of 1–2 or less), diffraction is important, and the received
intensity is a complex function of position, with no well-defined shadow region
[42]. When the obstacle subtends many Fresnel zones (>3–5), the tag lies in a
fairly well-defined shadow and cannot communicate with the reader. Fresnel
291
292
7 RFID
Figure 7.21 Tags in the shadow of a disk.
0.5 m
2m
m
5c
2m
diffraction theory or a numerical modeling tool calculate the attenuation due
to diffraction.
Example
A reader illuminates a 1-m diameter disk that is 2 m away (Figure 7.21). A tag is
placed behind the disk at (a) 5 cm and (b) 2 m. Determine which Fresnel zone
the edge of the disk lies at 915 MHz and comment on the impact of the tag
receiving the reader signal.
Solution
Use (5.31) assuming that the reader and tag are at the foci of the Fresnel
ellipsoid.
√
r2 (d + dr )
n𝜆dt dr
⇒n= n t
rn =
dt + dr
𝜆dt dr
𝜆=
3 × 1010
= 32.8 cm
915 × 106
502 (200 + 5)
= 15.6 Tag is in deep shadow and probably does not
32.8(200)(5)
detect signal.
502 (200 + 200)
(b) n =
= 0.76 Tag has good chance of detecting signal.
32.8(200)(200)
(a) n =
The backscattered power at the reader due to a semi-passive or chipless tag
is given by the radar range equation [41].
Preader =
Pt Greader Gtag c2 𝜎rcs
(4𝜋)2 fc2 r4
L
(7.7)
where 𝜎 rcs is the radar cross section (RCS) of the tag. A reasonable estimate
for our purposes assumes a modulated backscatter power level of 1/3 of the
absorbed power. Passive tag IC power requirements of tens or hundreds of
microwatts far exceed the receiver sensitivity. Figure 7.22 is the power received
at the reader when for a tag mounted on cardboard and aluminum at 915 MHz
7.6 Reader-Tag Communication
–40
Preader (dBm)
–60
Card
bo
ard
–80
Reader Sensitivity
–100
–120
Alum
inium
–140
1
2
4
r (m)
6
8
10
20
Figure 7.22 Power received at the reader when for a tag mounted on cardboard and
aluminum at 915 MHz
[41]. If the reader threshold is Preader = −80 dBm, then the tag mounted on cardboard has an IZ of about 5 m while the tag mounted on aluminum has an IZ less
than 1 m.
7.6.2.3
Chipless Tags
Chipless tags dramatically reduce the cost of a tag by removing the most
expensive part: the silicon IC [43]. Some chipless tags use plastic or conductive polymers instead of silicon-based microchips. Other chipless tags use
materials that backscatter the reader signal in a unique signature. Figure 7.23
lists the categories of chipless RFID tags. Companies are experimenting with
embedding RF reflecting fibers in paper to prevent unauthorized photocopying
of certain documents. Reflective tattoo inks identify farm animals.
Delay-line-based chipless tags have microstrip discontinuities after a delay
line [44]. A short interrogation pulse from a reader (1 ns) reflects from carefully placed discontinuities in the microstrip line and results in reflections as
shown in Figure 7.24. The length of the delay-line between the discontinuities
determines the time delay between the reflections.
One approach to a chipless tag places the RF circuit between receive and
transmit antennas [45]. The circuit has cascaded resonators (Figure 7.25) that
resonate at specific frequencies in order to create bandstop filters that cause
an attenuation and phase jump at these frequencies. The reader interprets the
spectrum of the tag signal as bits. The tag signal resonant frequencies correspond to a code. The chipless tag in Figure 7.26 consists of five second-order
Piano curves that produce five peaks in the RCS [46]. These peaks represent a
bit sequence.
293
7 RFID
Chipless RFID tags
Spectral
signature based
TDR based
Nonprintable
Printable
Chemical
TFTC
SAW tags
Delay-linebased tags
Amplitude/phase backscatter
modulation based
Left-hand (LH)
delay lines
Planar Circuits
Nanometric
materials
Capacitively
tuned dipoles
Stub-loaded
patch antenna
Ink-tattoo
chipless RFID
Space filling
curves
Remote complex
impedance
LC resonant
Carbon
nanotube loading
Multiresonator
based
Multiresonant
dipoles
Figure 7.23 Categories of chipless RFID tags. Source: Reprinted with permission of
Preradovic and Karmakar [43]. © 2010, IEEE.
Figure 7.24 Interrogation and
coding of delay-line-based
chipless tag. Source: Reprinted
with permission of Preradovic
and Karmakar [43]. © 2010, IEEE.
Amplitude
Input signal
011
Reflected signal
‶Generated ID: 011″
110
Reflected signal
‶Generated ID:110″
f1 – Δf
fn + Δf
Frequency
fn + Δf
Frequency
Interrogation signal
generated by reader
Rx Ant.
To reader
Tx Ant.
50-Ω line
Phase
From
reader
Chipless tag
Magnitude
0 1 0 1 0 1 0 1 Time
00 00 01 01 10 10 11 11
Magnitude
Pulse position
modulation code
representation
Phase
294
f1 – Δf
Spiral resonators—
resonating frequencies:
f1, f2, f3, …,fn
f1 f2 f3, …,fn
f1 – Δf
fn + Δf
Frequency
f1 f2 f3, …,fn
f1 – Δf
fn + Δf
Frequency
Received signal from tag
Figure 7.25 Diagram of the operation of a multiresonator-based chipless RFID tag. Source:
Reprinted with permission of Tedjini et al. 2013 [45]. © 2010, IEEE.
Problems
Figure 7.26 Five-bit
piano-curve-based tag and tag radar
cross section spectral signature.
Source: Reprinted with permission of
Preradovic and Karmakar [43]. ©
2010, IEEE.
y
x
–15
Ey
–20
RCS (dB)
–25
–30
–35
–40
–45
–50
0.5
Figure 7.27 Operating principle
of left-hand-delay-line based
chipless RFID tag. Source:
Reprinted with permission of
Preradovic and Karmakar [43]. ©
2010, IEEE.
0.6
0.7
0.8
0.9
Frequency (GHz)
Γ2
Γ1
φ1
ϑ1
e jφ1
Carrier phase
Carrier
envelope
T · e jφ0
Γ3
φ2
ϑ2
T · e jφ0
1
Reflection
section
φ3
ϑ3
Delay line
section
φ3 + 2(ϑ1 + ϑ2)+6φ0
φ1 + 2φ0
φ2 + 2ϑ1 + 4φ0
t
Figure 7.27 shows the operation of a left-hand (LH) delay line chipless
RFID tag [47]. The interrogation pulse propagates through the periodic LH
delay lines. The backscattered pulse from the discontinuities is coded with the
reflected signal phase relative to a reference phase. The reflected signals have
equal amplitude envelopes but their phases due to Γ1 , Γ2 , and Γ3 are 𝜙1 , 𝜙2 ,
and 𝜙3 . This approach encodes data with a higher order modulation scheme
(e.g. QPSK) that increases throughput at the expense of a higher SNR.
Problems
7.1
Assume the 433 MHz tag IC has a threshold power level of −10 dBm. The
reader transmits 30 dBm through an antenna with 6 dB gain. Assume the
tag antenna is isotropic and 𝜂 PCE = 0.3. What is the maximum extent of
the IZ?
295
296
7 RFID
7.2
The barcode data is 001234567890x where x is the checksum digit.
Find x.
7.3
What is the range of the power conversion efficiency for a Schottky diode
Dickson multiplier that has four stages?
7.4
Assume the 865 MHz reader has a threshold power level of −30 dBm.
The reader antenna has a 6 dB gain. The tag antenna has a gain of 0 dB.
Let L = 1.
7.5
Plot the PIE coding for the bits [1 1 0 1].
7.6
Plot the FM0 coding for the bits [1 1 0 1].
7.7
A binary data stream has the bits [1 1 0 0]. Plot the baseband symbol,
square wave modulator, and Miller encoding with M = 2.
7.8
If f c = 125 kHz and the tag bit rate is f c /16, then find the data rate and
the time needed to receive 128 bits.
7.9
Find the reflection coefficients for a 1 and 0 bit when Zant = 45 + j20 Ω,
ZL0 = 50 Ω, and ZL1 = 10 Ω.
7.10
If the tag must receive −10 dBm, then design a reader give a tag with
Gtag = 0 dB, 𝛿 p = 0.5, T load = 0.5, 𝜂 ob = 0.5, Lblock = 1.0, and F tag = 1.0.
Assume that the frequency is 3 GHz and the IZ extends to 10 m.
7.11
What fraction of the power gets delivered to the IC if Zant = 45 + j20 Ω,
ZL0 = 50 Ω, and ZL1 = 10 Ω.
References
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2 Want, R. (2006). An introduction to RFID technology. IEEE Pervasive Com-
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References
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Patent 2,612,994, 7 October 1952.
7 https://electronics.howstuffworks.com/gadgets/high-tech-gadgets/upc.htm
(accessed 20 July 2018).
8 Cardullo, M.W. and Parks, W.L. (1973). Transponder apparatus and system.
US Patent 3,713,148, 23 January 1973.
9 Roberti, M. (2005). The history of RFID technology. RFID J., http://www
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11 Lumpkins, W. (2015). RFID: an evolution of change, from World War II to
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14 https://www.rfidjournal.com/glossary/?T (accessed 8 June 2018).
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16 http://rfid4u.com/rfid-basics-resources/how-to-select-a-correct-tag-
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20 WEBINAR SERIES #4 EPC/RFID STANDARDS AND RFID - STUFF YOU
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22
23
24
25
26
NEeD TO KNOW," webinar presentation by GS1 US, 2012. (accessed 28
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Shretha, S., Vemagiri, J., Agarwal, M., and Varahramyan, K. (2007). Transmission line reflection and delay-based ID generation scheme for RFID and
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Tedjini, S., Karmakar, N., Perret, E. et al. (2013). Hold the chips: chipless
technology, an alternative technique for RFID. IEEE Microwave Mag. 14 (5):
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46 McVay, J., Hoorfar, A., and Engheta, N. (2006). Space-filling curve RFID
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(17–19 January 2006), pp. 199–202.
47 Caloz, C. and Itoh, T. (2004). Transmission line approach of left-handed
(LH) materials and microstrip implementation of an artificial LH transmission line. IEEE Trans. Antennas Propag. 52 (5): 1159–1166.
299
301
8
Direction Finding
Direction finding (DF), also known as angle of arrival (AOA) or direction
of arrival (DOA) estimation, locates radio frequency (RF) signals in the
environment. Pointing the antenna main beam in the direction of a signal
produces the highest received signal level but has limited resolution due to the
antenna beamwidth which is inversely proportional to aperture size. Strong
signals entering the sidelobes overwhelm a weaker signal received by the main
beam, leading to poor AOA estimates. Main beams slowly decrease away
from the peak, so small angle changes in the signal produce small changes in
the antenna output. For instance, Figure 8.1 shows that the antenna pattern
changes by 3 dB over an 11.8∘ angular range. This range typically defines an
antenna’s resolution (3 dB beamwidth).
Consequently, most approaches to DF locate signals using nulls rather than
peaks in the antenna patterns. Nulls have steep sides, so small changes in angle
produce large changes in output power that a receiver easily detects. A signal
entering a null produces zero antenna output. The angular change of the difference pattern null in Figure 8.1 for a 3 dB change in antenna pattern is about
0.4∘ . At −20 dB, the null width is 2.0∘ . Compare that null angular accuracy with
the 12.8∘ beamwidth of the sum pattern.
This chapter starts by presenting some relatively simple approaches to DF
that use the main beam or one null. The rest of the chapter covers sophisticated
digital signal processing algorithms that precisely locate multiple signals using
nulls.
8.1 Direction Finding with a Main Beam
An antenna has a maximum output power when the antenna main beam points
directly at the signal. The elevation and azimuth of the antenna pointing direction determines the angular location of the signal.
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
8 Direction Finding
Figure 8.1 Antenna pattern null
vs. peak for DF.
Sum pattern
0
3 dB
12.8°
–5
AF (dB)
302
–10
Difference
pattern
–15
2.0°
3 dB
–20
–10
8.1.1
–5
0
θ (deg)
5
10
Array Output Power
The power received by an antenna equals the signal power times the antenna
gain in the direction of the signal. Maximum antenna output occurs when the
signal enters the antenna’s main beam. Sidelobes and nulls significantly attenuate signals incident on them. The array output of an N element linear array
(Figure 8.2) with element spacing d due to M signals incident on the array from
M angles at 𝜃 m is given by [1]
sout =
M N
∑
∑
wn (sm ejk(n−1)d sin 𝜃m + 𝜈n ) = wT (As + n)
(8.1)
m=1 n=1
where
s = [s1 s2 · · · sM ]T = signal amplitude vector
[
]T
w = w1 w2 · · · wN = element weight vector
⎡1
⎢ejkd sin 𝜃1
A=⎢
⎢⋮
⎢ jk(N−1)d sin 𝜃
1
⎣e
1
e
jkd sin 𝜃2
⋮
ejk(N−1)d sin 𝜃2
··· 1
⎤
⎥
⎥
⎥
⋱ ⋮
⎥
jk(N−1)d sin 𝜃M ⎦
··· e
· · · ejkd sin 𝜃M
= array steering matrix
[
]T
n = 𝜈1 𝜈2 · · · 𝜈N = element noise vector
The signal amplitude vector contains the relative amplitudes of the M signals
incident on the array. Column m in the array steering matrix contain the phase
differences at the elements due to a signal incident from 𝜃 m . Each element has
2
,
independent random noise with a complex value 𝜈 n and average power 𝜎noise
assuming additive white Gaussian noise (AWGN).
8.1 Direction Finding with a Main Beam
z
Figure 8.2 Diagram of a DF
array.
θ
S2
S1
S3
• • •
SM
x
w1
w2
wN
• • •
∑
The average array output power is proportional to the expected value of the
amplitude of the array output signal squared:
P = E[s† s]
= E[|wT (As + n)|2 ]
= E[w† (As + n)(As + n)† w]
= w† Cw
(8.2)
where E[•] is the expected value and “†” is the complex conjugate transpose.
The covariance matrix, C, is defined by
C = E[(As + n)(As + n)† ]
= E[Ass† A† ] + E[ns† A† ] + E[Asn† ] + E[nn† ]
= AE[ss† ]A† + E[ns† ]A† + AE[sn† ] + E[nn† ]
= Cs + Cs−noise + Cnoise−s + Cnoise
(8.3)
and
Cs = signal covariance matrix
Cs−noise = signal–noise covariance matrix
Cnoise−s = noise–signal covariance matrix
Cnoise = noise covariance matrix.
The expected value operator applies to time-varying quantities, so the elements
of the covariance matrix describe the cross-correlation between the signals at
all of the elements. For instance, row m and column n describes how close the
signal at element m looks like the signal at element n.
303
304
8 Direction Finding
Uncorrelated signal and noise have Cs−noise = 0 and Cnoise−s = 0 leaving only
the signal and noise covariance matrixes
C = Cs + Cnoise
(8.4)
2
The noise covariance matrix diagonal elements equal the noise variance, 𝜎noise
,
and the off-diagonal elements equal zero, because the noise at one element is
uncorrelated with the noise at another element
Cnoise
⎡𝜎 2
0
0 ⎤
⎥
⎢ noise
⋱
0 ⎥
=⎢ 0
⎥
⎢ 0
2
0 𝜎noise
⎦
⎣
(8.5)
When no noise is present, then C = Cs . When no signal is present, then
C = Cnoise .
8.1.2
Periodogram
Scanning the antenna main beam maps the signal locations by recording output
power as a function of angle. A plot of the output power vs. angle obtained from
(8.2) is called a periodogram [2]. A periodogram due to one signal is the same as
the antenna pattern. Periodograms have three problems with locating signals:
1. Resolution: The antenna beamwidth is inversely proportional to the aperture
size in wavelengths. Two or more signals within the main beam beamwidth
appear as one signal – they cannot be resolved. Consequently, fine resolution
requires a large electrical aperture.
2. Sidelobes: Strong signals entering a sidelobe cannot be distinguished from
weaker signals entering the main beam in the periodogram. Low sidelobes
mitigate this problem but produce a wider main beam, hence degraded resolution.
3. Accuracy: As noted in Figure 8.1, the error in the angular location of a signal
detected in a main beam is approximately ±𝜃 3dB /2.
The following example highlights these problems and sets the stage for digital
beamforming DF.
Example
Plot the periodogram of an eight-element uniform linear array of isotropic elements with d = 𝜆/2 spacing when signals of equal power are incident at 𝜃 1 = 30∘ ,
10∘ , 0∘ , and − 60∘ . Ignore noise.
Solution
The output power is given by
P(𝜃) = w† Cs w
(8.6)
8.1 Direction Finding with a Main Beam
Figure 8.3 Polar plot of the periodogram of an
eight-element uniform array when four equal
power signals are incident at 𝜃 1 = 30∘ , 10∘ , 0∘ ,
and − 60∘ .
90
60
S1
0 dB
30
–10 dB
S2
0
S3
–30
–60
–90
S4
In order to steer the beam to 𝜃 s , the weights, w, must include a steering phase.
Since the array is uniform, the weights are given by
wn = e−jk(n−1)d sin 𝜃s
(8.7)
Figure 8.3 shows a polar plot of (8.6) for 0 ≤ 𝜃 ≤ 180∘ . The periodogram has
peaks in the directions of −60∘ and 30∘ , so these signals are resolved. A single
peak at 5∘ is due to the two signals at 0∘ and 10∘ . Since they are separated by
less than a beamwidth, they cannot be resolved. Note that a weak response
also appears at 90∘ even though no signal is present. Signals entering sidelobes
create that ghost signal when the array main beam is swept toward 90∘ .
8.1.3
Wullenweber Array
A circular array of N elements scans 360∘ in azimuth by having an N a subset of contiguous elements active at a time. Figure 8.4 shows the main beam
pointing normal to the center of the N a active elements connected to the feed
network. The remaining N − N a elements have no connection to the feed network. To steer the beam from active elements 1 to N a by 360∘ /N, element 1
is disconnected from the feed network and element N a + 1 is connected to the
feed network. Performing this switching N times completes a 360∘ azimuth
scan of the main beam.
If the circular array lies in the x–y plane, then its array factor is
AF =
Na
∑
n=1
wn ejkrc cos(𝜙−𝜙n )
(8.8)
305
306
8 Direction Finding
y
Figure 8.4 Diagram of a circular array.
Feed
𝜙
x
Elements
where
rc = radius of circular array
𝜙n = angular location of element n.
Since the array lies on a curved surface, the elements need a nonlinear phase
compensation to form a coherent main beam in the beam pointing direction 𝜙s .
wn = e−krc cos(𝜙s −𝜙n )
(8.9)
Hans Rindfleisch invented the circular Wullenweber array during WW II for
HF DF [3]. Figure 8.5 shows an example of an AN/FRD-10 Circularly Disposed
Antenna Array (CDAA) at Gandor, NL, Canada.
Figure 8.5 Wullenweber array near
Gandor, NL, Canada. Source: Imagery
©2018 DigitalGlobe. Map data ©2018
Google.
Reflector
Elements
Ground screen
8.2 Direction Finding with a Null
8.2 Direction Finding with a Null
As already shown, a null has greater angular accuracy than a main beam, so
pointing a null at a signal until the output goes to zero accurately locates the signal. A loop antenna has a null perpendicular to the plane of the loop (Chapter 4).
The HF DF loop antenna in Figure 8.6 has a motorized base that mechanically
steers the antenna in azimuth. Even small antennas have sharp nulls for precisely locating a signal.
An array difference pattern has a sharp null at boresight rather than a peak.
Steering this null in the direction of a signal eliminates the contribution of that
signal to the array output. A uniform difference pattern results when half the
elements have a 180∘ phase shift or wn = 1 for n ≤ N/2 and wn = − 1 for n > N/2,
assuming N is even. A closed form expression for a uniform difference array
with an even number of elements is given by
( )
1 − cos N𝜓
2
AF =
( )
j sin 𝜓2
Figure 8.6 HF DF loop antenna.
(8.10)
307
8 Direction Finding
θ3dB
Figure 8.7 Uniform sum
and difference patterns.
0
Sum
Difference
–5
AF (dB)
308
–10
–15
–20
–60
–40
–20
0
θ (deg)
20
40
60
Bayliss found a low sidelobe taper for difference patterns that resembles the
Taylor taper for sum patterns [4] in which n − 1 sidelobes of equal height next
to the main beam, while the rest decrease away from boresight. Simultaneous
sum and difference patterns in a monopulse radar enable detection and location
of targets.
Example
Plot the sum and difference patterns for an eight-element uniform array with
𝜆/2 spacing.
Solution
The sum pattern weights are
[
]
w= 1 1 1 1 1 1 1 1
and the difference weights are
[
]
w = 1 1 1 1 −1 −1 −1 −1
Substitute these weights into the array factor formula to calculate the array factors. Figure 8.7 shows the two patterns that are normalized to the peak of the
sum pattern (N). The sum pattern 3 dB beamwidth is 12.8∘ . Note how narrow
the null appears compared to the main beam. A small angular change near the
null of the difference pattern produces a large change in the output power. This
approach minimizes the output power to determine the signal location rather
than the periodogram that attempts to maximize the signal output power.
8.3 Adcock Arrays
The original Adcock array had four monopoles on the corners of a square
(Figure 8.8) [5]. The elements along the x-axis combine out of phase so that
8.3 Adcock Arrays
Figure 8.8 Diagram of a four monopole Adcock array.
0,
–
d
2
d
d ,0
2
2
,0
z
θ
0, –
d
2
y
ϕ
x
the resulting array factor has peaks at 𝜙 equals 0∘ and 180∘ and nulls at 90∘
and 270∘ . The x-axis array factor is written as
(
)
d
AFx (𝜃, 𝜙) = 2j sin k sin 𝜃 cos 𝜙
(8.11)
2
In contrast, the elements along the y-axis combine out of phase and in array
factor peaks at 90∘ and 270∘ and nulls at 0∘ and 180∘ . The y-axis array factor is
given by
(
)
d
AFy (𝜃, 𝜙) = 2j sin k sin 𝜃 sin 𝜙
(8.12)
2
where d is the separation between the elements along the x- and y-axis. AFx and
AFy provide an estimate of the azimuth and elevation angles of a signal arriving
from (𝜃 s , 𝜙s ) [6].
)
(
sin k d2 sin 𝜃s sin 𝜙s
AFy (𝜃s , 𝜙s )
(8.13)
tan 𝜙s ≈
=
)
(
AFx (𝜃s , 𝜙s ) sin k d sin 𝜃 cos 𝜙
s
s
2
√
1 4
2
cos 𝜃s ≈
AFx (𝜃s , 𝜙s ) + AF2y (𝜃s , 𝜙s )
(8.14)
kd
A circular Adcock array provides better estimates for the AOA by placing
additional element pairs on opposite sides of a circle with a center at the origin
of the x–y axes [2]. Figure 8.9 shows an example of an eight-element Adcock
Figure 8.9 Diagram of an eight monopole
circular Adcock array.
z
y
θ
ϕ
x
309
8 Direction Finding
Figure 8.10 The array factors for the eight-element
Adcock array.
90°
180°
0°
270°
Figure 8.11 A graph of the calculated vs.
estimated azimuth angles for the
eight-element Adcock array.
360
Exact
Estimate
Estimated ϕ (deg)
310
270
180
90
0
270
90
180
Actual ϕ (deg)
360
array. The outputs of two adjacent elements add to get a single output. These
four outputs are then combined using (8.11) and (8.12). If the circle has a radius
of 0.25𝜆 with eight elements, then the two array factors appear in Figure 8.10
and the azimuth angle estimate is shown in Figure 8.11. This small radius produces a small error in the AOA estimate. Increasing the radius in terms of
wavelength also increases the error in the AOA estimate.
8.4 Eigenbeams
Since the covariance matrix in (8.4) describes the relationship between the signals at all the elements, its eigenvectors and eigenvalues contain important
signal information. An eigen-decomposition of the covariance matrix in (8.4)
results in
C = Q𝚲𝜆 Q−1 =
M
∑
2
𝜆m Q[∶, m]Q† [∶, m] + 𝜎noise
IN
m=1
⏟⏟⏟
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
CN
Cs
(8.15)
8.4 Eigenbeams
where
Q = N × N matrix whose columns are the eigenvectors
⎡𝜆1 0 · · · 0 ⎤
⎢0 𝜆 ⋱ ⋮ ⎥
2
⎥
𝚲𝜆 = ⎢
⎢⋮ ⋱ ⋱ 0 ⎥
⎥
⎢
⎣ 0 · · · 0 𝜆N ⎦
= N × N matrix whose diagonal are the N eigenvalues
𝜆n = eigenvalue associated with the eigenvector in column n
IN = N × N identity matrix
M uncorrelated signals incident on an N element array (N > M) produces
M signal eigenvalues and N − M noise eigenvalues. Large eigenvalues (𝜆n ≫
2
2
) correspond to signals while small eigenvalues (𝜆n ≈ 𝜎noise
) correspond
𝜎noise
to noise. Noise eigenvalues are independent of 𝜃 and approximately equal the
noise variance. Signal eigenvalues, on the other hand, are a function of signal
direction and are proportional to the power of the signals.
Let eigenvector n from Q equal the array weight vector w = Q[:, n]. Substitute
w into the equation for the array factor from Chapter 4 to get eigenbeam n [7]
EBn (𝜃) =
N
∑
Q[∶, n]ejk(n−1)d sin 𝜃
(8.16)
n=1
Figure 8.12 graphs the 20 eigenbeams associated with a 20-element linear array
(𝜆/2 spacing) and two signals having the same normalized power of 1 W inci2
= 0.09 W. The signal eigenbeams have
dent at 𝜃 = − 50∘ and 𝜃 = 30∘ with 𝜎noise
main beams that point in the directions of the signals (Figure 8.12a). Note that
the two signal eigenvectors differ and produce unique eigenbeams. All 18 noise
eigenbeams have nulls pointing at 𝜃 = − 50∘ and 𝜃 = 30∘ (Figure 8.12b). Any
–50°
30°
0
AF (dB)
AF (dB)
0
–25
–50
–90
–45
0
θ (deg)
(a)
45
90
–50°
30°
–25
–50
–90
–45
0
θ (deg)
45
90
(b)
Figure 8.12 Eigenbeams for a 20-element array with two equal power signals incident at
𝜃 = − 50∘ and 𝜃 = 30∘ . (a) Signal eigenbeams and (b) Noise eigenbeams.
311
312
8 Direction Finding
of the 20 eigenvectors have main beams or nulls pointing in the directions of
signals. The signal eigenbeams use eigenbeam maxima, while the noise eigenbeams use eigenbeam minima.
Example
Plot the eigenbeams of a four-element uniform linear array with 𝜆/2 spacing
2
= 0.16 W∕m2 .
when a 1 V/m plane wave is incident at 30∘ and 𝜎noise
Solution
First form the covariance matrix:
0.4957
0.0061 + j0.3316
−0.3246
0.0015 − j0.3316⎤
⎡
⎢0.0061 − j0.3316
⎥
0.4953
0.0025 + j0.3316
−0.3289
⎥
C=⎢
⎢
−0.3246
0.0025 − j0.3316
0.4865
0.0022 + j0.3316⎥
⎢
⎥
−0.3289
0.0022 − j0.3316
0.4870
⎣0.0015 + j0.3316
⎦
Next, find the eigenvectors and eigenvalues of the covariance matrix using
MATLAB.
[eigvec,eigval]=eig(C)
The normalized amplitudes and phases of the eigenvectors along with the
eigenvalues appear in Table 8.1. The signal eigenvalue (𝜆1 = 1.52) is easy to
identify, because it is much greater than the noise eigenvalues. The signal
Table 8.1 Eigenvectors and eigenvalues for a four-element uniform array with a 1 V/m plane
wave incident at −50∘ and a 1 V/m plane wave incident at 30∘ .
Eigenvectors
Signal
Noise
Q[:, 1]
Q[:, 2]
Q[:, 3]
Q[:, 4]
|Q[:, 1]|
∠Q[:, 1]
|Q[:, 2]|
∠Q[:, 2]
|Q[:, 3]|
∠Q[:, 3]
|Q[:, 4]|
∠Q[:, 4]
0.9964
0∘
90∘
0.4272
0∘
0.7380
0∘
1.0000
0∘
0.3754
0.5372
169.23∘
322.34∘
0.7576
0.9469
78.41∘
176.65∘
1.0000
180∘
270∘
0.7216
21.89∘
18.22∘
1.0000
271.41∘
0.6712
190.87∘
0.4911
−4.60∘
1.0000
0.9877
0.9922
Eigenvalues
Signal
𝝀1
1.52
Noise
𝝀2
𝝀3
𝝀4
0.22
0.20
0.19
8.5 Direction Finding Algorithms
0
0
30°
AF (dB)
AF (dB)
30°
–25
–50
–90
–45
0
θ (deg)
(a)
45
90
–25
–50
–90
–45
0
θ (deg)
(b)
45
90
Figure 8.13 Plot of eigenbeams for the four-element uniform array with 1 V/m plane wave
incident at 30∘ . (a) Signal eigenbeams and (b) Noise eigenbeams.
eigenbeam associated with the array weights Λ1 has a peak at 30∘ due to the
linear progressive phase shift between elements. Setting the steering phase
equal to the phase difference between adjacent elements gives a beam pointing
direction of
( ∘ )
360∘ 𝜆
90
= 30∘
sin 𝜃s ⇒ 𝜃s = sin−1
𝛿s = kd sin 𝜃s =
𝜆 2
180∘
Figure 8.13 is a plot of the four eigenbeams. The three noise eigenbeams have
nulls at 30∘ .
8.5 Direction Finding Algorithms
As noted in the previous example, the covariance matrix contains information
about the direction and relative power of the signals incident on the array.
An eigenanalysis of the covariance matrix reveals the signal locations and
strengths. This section explains how to exploit the covariance matrix to find
the location of signals.
8.5.1
Capon’s Minimum Variance
Capon’s method (also called minimum variance spectral estimation or the maximum likelihood method) calculates an estimate of the signal power as a function of angle [8]. This approach minimizes the output power while forcing the
signal power to remain constant:
Minimize ∶ w† Cw
Subject to ∶ wT As = 1
(8.17)
Thus, the output power is minimized, except in the direction of the received
signals. The analytical solution to (8.17) is
w=
C−1 A
A† C−1 A
(8.18)
313
314
8 Direction Finding
which produces the spectrum given by
P(𝜃) =
1
(8.19)
A† (𝜃)C−1 A(𝜃)
Capon’s method does not work well with correlated signals, because they might
add to zero and avoid detection. It also requires computing the covariance
matrix inverse which may be slow and error prone.
Example
An eight-element uniform array with 𝜆/2 spacing has signals with amplitude
1.0 V/m incident at 𝜃 1 = 30∘ , 10∘ , 0∘ , and − 60∘ . Find the Capon spectrum when
2
= 0.1 W∕m2 .
𝜎noise
Solution
The covariance matrix is calculated to be
3.91 − j0.00
0.98 − j1.07
1.06 − j1.65
0.59 + j0.85
1.28 − j1.74
0.50 − j0.68
−1.78 − j0.52
0.99 + j1.64
0.98 + j1.07
4.20 + j0.00
0.96 − j1.08
1.28 − j1.65
0.72 + j1.03
1.38 − j1.92
0.66 − j0.51
−1.80 − j0.50
1.06 + j1.65
0.96 + j1.08
4.11 + j0.00
1.03 − j1.23
1.17 − j1.69
0.59 + j0.70
1.40 − j1.98
0.54 − j0.74
0.59 − j0.85
1.28 + j1.65
1.03 + j1.23
4.19 − j0.00
1.07 − j0.99
1.15 − j1.65
0.81 + j0.99
1.27 − j1.77
1.28 + j1.74
0.72 − j1.03
1.17 + j1.70
1.07 + j0.99
4.21 + j0.00
0.95 − j1.28
1.30 − j1.60
0.66 + j0.88
0.50 + j0.68
1.38 + j1.92
0.59 − j0.70
1.15 + j1.65
0.95 + j1.28
3.98 − j0.00
0.98 − j1.01
0.97 − j1.60
−1.78 + j0.52
0.66 + j0.51
1.40 + j1.98
0.81 − j0.99
1.30 + j1.60
0.98 + j1.01
4.28 − j0.00
1.03 − j1.09
0.99 − j1.64
−1.80 + j0.50
0.54 + j0.74
1.27 + j1.77
0.66 − j0.88
0.97 + j1.60
1.03 + j1.09
3.89 + j0.00
and the array steering matrix is
⎡−1.00 + j0.10
⎢
⎢ 0.87 − j0.50
⎢−0.59 + j0.81
⎢
⎢ 0.21 − j0.98
A=⎢
⎢ 0.21 + j0.98
⎢−0.59 − j0.81
⎢
⎢ 0.87 + j0.50
⎢ −1.0 − j0.10
⎣
1.00 + j0.00 −0.33 + j0.94 0.71 − 0.71⎤
⎥
1.00 + j0.00 0.21 + j0.98 −0.71 − 0.71⎥
1.00 + j0.00 0.68 + j0.73 −0.71 + 0.71⎥
⎥
1.00 + j0.00 0.96 + j0.27 0.71 + 0.71⎥
⎥
1.00 + j0.00 0.96 − j0.27 0.71 − 0.71⎥
1.00 + j0.00 0.68 − j0.73 −0.71 − 0.71⎥
⎥
1.00 + j0.00 0.21 − j0.98 −0.71 + 0.71⎥
1.00 + j0.00 −0.33 − j0.94 0.70 + 0.71⎥⎦
Substitute the inverse of C and A into (8.19) and plot the result. In terms of
MATLAB, the key command is
Pc(ic)=1/abs(A(:,ic).'* inv(C) *conj(A(:,ic)));
where ic indicates an angle. Figure 8.14 shows that Capon’s spectrum has sharp
lines at the signal angles. Unlike the periodogram, it distinctly separates the
signals at 0∘ and 10∘ .
8.5 Direction Finding Algorithms
0
Pcap (dB)
Figure 8.14 Plot of the Capon
power spectrum of an eight-element
array with four signals present.
–25
–90
8.5.2
–45
0
θ (deg)
45
90
Pisarenko Harmonic Decomposition
Pisarenko harmonic decomposition (PHD) finds the power spectrum due to M
sinusoidal signals incident on an array with AWGN present. As seen from the
eigenbeam example, the noise eigenvectors approximately equal the noise vari2
)
ance. The eigenvector corresponding to the smallest eigenvalue (𝜆min ≈ 𝜎noise
minimizes the mean squared error of the array output with the constraint that
the norm of the weight vector equals one. In this case, the power spectrum is [9]
P(𝜃) =
1
|A† (𝜃)Q[∶, min]|2
(8.20)
where Q[:, min] is the eigenvector that corresponds to the minimum eigenvalue
𝜆min . Since the noise eigenvector is orthogonal to all the signal eigenvectors,
the denominator of (8.20) goes to zero in the directions of all signals, producing peaks in the spectrum. PHD serves as a starting point for more sophisticated approaches that are less sensitive to noise. An interesting note is that
(8.20) is actually the inverse of the eigenbeam power pattern corresponding to
𝜆min . In other words, (8.20) is the magnitude squared of the upside down array
factor.
Example
An eight element uniform array with 𝜆/2 spacing has signals with amplitude
1.0 V/m incident at 𝜃 1 = 30∘ , 10∘ , 0∘ , and − 60∘ . Find the PHD spectrum when
2
= 0.1 W∕m2 .
𝜎noise
Solution
In order to make use of (8.20), the eigenvectors and eigenvalues of the covariance matrix must be found. The MATLAB commands are
[eigvec,eigval]=eig(C)
Pphd(ic)=1/abs(A(:,ic).'*eigvec(:,ii))ˆ2
315
8 Direction Finding
Figure 8.15 Plot of the PHD power
spectrum of an eight-element array
with four signals present.
0
PPHD (dB)
316
–25
–50
–90
–45
0
θ (deg)
45
90
where ii points to the eigenvector corresponding to the minimum eigenvalue.
Figure 8.15 shows the very sharp lines in the PHD spectrum at the signal angles.
It easily separates the signals at 0∘ and 10∘ . If the reader looks at Figure 8.15
upside down, it looks like an array factor with peaks at −90∘ and 53∘ and nulls
in the directions of the signals!
8.5.3
MUSIC Algorithm
MUSIC (MUltiple SIgnal Classification) resembles PHD but replaces the minimum eigenvector in the power spectrum estimate with an average of the noise
eigenvectors of the covariance matrix [10]. Since all of the noise eigenbeams
have nulls in the direction of the signals, any of them work for DF. Averaging
them makes MUSIC more robust to noise compared to PHD that only uses one
of them. The MUSIC spectrum is given by
P(𝜃) =
1
|A† (𝜃)Q[∶, M
+ 1 ∶ N]|2
(8.21)
where Q[:, M + 1 : N] are the eigenvectors corresponding to the noise (eigen2
values equal 𝜎noise
). MUSIC requires uncorrelated or at most mildly correlated
signals. The MUSIC algorithm accurately estimates the number and strengths
of signals as well as their AOA for a calibrated array and uncorrelated signals
[11].
In practice, no exact demarcation between signal and noise eigenvectors
exists. For an unknown number of signals, the eigenvalues only estimate the
number of signals present. An actual antenna array generates an approximation to the real covariance matrix. The noise eigenvalues associated with this
approximate covariance matrix are close in value but not equal. The ratio of the
geometric mean to their arithmetic mean of the noise eigenvalues measures
the closest of the eigenvalues.
Example
An eight-element uniform array with 𝜆/2 spacing has signals with amplitude
1.0 V/m incident at 𝜃 1 = 30∘ , 10∘ , 0∘ , and − 60∘ . Find the MUSIC spectrum
2
= 0.1 W∕m2 .
when 𝜎noise
8.5 Direction Finding Algorithms
0
Pmu (dB)
Figure 8.16 Plot of the MUSIC
power spectrum of an
eight-element array with four
signals present.
–25
–50
–90
–45
0
θ (deg)
45
90
Solution
The MUSIC spectrum in Figure 8.16 is similar to the corresponding PSD spectrum.
8.5.4
Root MUSIC
Capon’s method, PHD, and MUSIC estimate the power spectrum. Peaks in the
spectrum correspond to signals and their locations. A more practical algorithm
called root-MUSIC, estimates the AOAs based on the roots of the array polynomial rather than the power spectrum [12]. Taking the z-transform of the
denominator of (8.21) results in
∑∑
N−1 N−1
A† (𝜃)Q[∶, M + 1 ∶ N]Q† [∶, M + 1 ∶ N]A(𝜃) =
zn Cmn z−m
m=0 n=0
∑
2N−2
=
𝓁=0
c𝓁 z 𝓁
(8.22)
where
sin 𝜃
z = ejknd
∑
c𝓁 = n−m = 𝓁 C mn = sum of 𝓁th diagonal elements of matrix Q[:, M + 1 : N]Q†
[:, M + 1 : N].
∗
The polynomial roots come in pairs: zm and 1∕zm
. One root lies inside the unit
circle while the other lies outside the unit circle. Both roots have the same
phase information which corresponds to the AOA. Roots on the unit circle
∗
and correspond to signals. Double roots
are double roots because zm = 1∕zm
result, because (8.22) is a power pattern. The power pattern equals the amplitude squared of the array factor. An array polynomial with N − 1 roots has a
power pattern polynomial with 2(N − 1) roots.
Of the 2N − 2 roots of the power pattern, only roots on or very close to the
unit circle correspond to the poles of the MUSIC spectrum and indicate the
presence of signals. The phase of the roots of the polynomial (AOAs) in (8.22)
are found from
(
)
arg(zm )
−1
(8.23)
𝜃m = sin
kd
317
318
8 Direction Finding
3 π/2
2
π
0
The algorithm ignores spurious roots not close
to the unit circle. Root MUSIC outperforms
MUSIC in a low signal to noise ratio (SNR) environment.
1
4
Example
An eight element uniform array with 𝜆/2 spacing
has
signals with amplitude 1.0 V/m incident at
Figure 8.17 Unit circle
representation of all the roots
𝜃 1 = 30∘ , 10∘ , 0∘ , and − 60∘ . Find the location of
found using root MUSIC. Roots the signals using the root MUSIC algorithm when
corresponding to signals are
2
= 0.1 W∕m2 .
𝜎noise
–π/2
numbered in boxes.
Solution
The coefficients in (8.22) are
c𝓁 = −0.14 − j0.24, 0.33 − j0.11, −0.22 − j0.13, −0.45 + j0.52, −0.13
− j0.78, −0.62 + j1.18, −0.77 + j0.95 4.00, −0.77 − j0.95,
− 0.62 − j1.18, −0.13 + j0.78, −0.45 − j0.52, −0.22 + j0.13, 0.33
+ j0.11, −0.14 + j0.24
The 2N − 2 roots of the array polynomial in (8.22) are listed in Table 8.2. A unit
circle plot of the roots appears in Figure 8.17. Signals correspond to the double
roots that have the same phase and are near the unit circle. One of the roots is
outside and one inside the unit circle. The AOAs are found from the phases via
(𝜓 )
(𝜓 )
m
m
= sin−1
= −60.02∘ , 0.06∘ , 9.99∘ , 29.9∘
(8.24)
𝜃m = sin−1
kd
𝜋
The AOAs are slightly off due to the noise.
8.5.5
Maximum Entropy Method
The maximum entropy method (MEM), also known as the all poles model or
the autoregressive model, uses the poles of the rational function model given
by [13, 14]
P(𝜃) =
1
|A† (𝜃)C−1 [∶, n]|2
(8.25)
where C−1 (:, n) is the nth column of the inverse covariance matrix. The selection
of n gives slightly different results.
Example
An eight-element uniform array with 𝜆/2 spacing has signals with amplitude
1.0 V/m incident at 𝜃 1 = 30∘ , 10∘ , 0∘ , and − 60∘ . Find the MEM spectrum when
2
= 0.1 W∕m2 .
𝜎noise
8.5 Direction Finding Algorithms
Table 8.2 Root MUSIC roots for the eight-element array with
four signals incident.
Root
Magnitude
Phase
1
2.04
No
2
1.77
−103.5∘
−69.1∘
3
1.61
No
4
1.01
147.4∘
31.2∘
5
0.99
6
1.01
7
0.99
8
1.01
9
0.99
10
1.01
11
0.99
12
0.62
13
0.56
14
0.49
Signal
No
Yes
31.2∘
89.8∘
Yes
89.8∘
−155.9∘
Yes
−155.9∘
0.2∘
0.2∘
147.4∘
Yes
−69.1∘
−103.5∘
No
No
No
Roots near the unit circle correspond to signals and come in pairs
at the same angle (one inside and one outside the unit circle).
0
PMEM (dB)
Figure 8.18 Plot of the MEM power
spectrum of an eight-element array
with four signals present.
–25
–50
–90
–45
0
θ (deg)
45
90
Solution
The resulting MEM spectrum is shown in Figure 8.18.
8.5.6
ESPRIT
ESPRIT (Estimation of Signal Parameters via Rotational Invariance
Techniques) applies stereoscopy to AOA estimation [15]. The algorithm
begins by dividing an N-element uniform linear array into two overlapping
subarrays, each having N − 1 elements and sharing N − 2 elements as shown
in Figure 8.19. The shared elements are called matched pairs. Each subarray
319
320
8 Direction Finding
Figure 8.19 An ESPRIT array has two
overlapping subarrays.
Shared Elements
1
2
Subarray 1
N –1
n
N
Subarray 2
has one element that the other subarray does not have. The M signals at the
elements in the two subarrays are written as
Subarray 1 ∶ X1 = A[1 ∶ N − 1, ∶]s + n
Subarray 2 ∶ X2 = A[2 ∶ N, ∶]s + n = A[1 ∶ N − 1, ∶]𝚽s s + n
(8.26)
where the M × M diagonal matrix, 𝚽s , is given by
jkd sin 𝜃1
0
···
0 ⎤ ⎡z1 0 · · · 0 ⎤
⎡e
⎥ ⎢
⎥
⎢
ejkd sin 𝜃2 0
0 ⎥ ⎢ 0 z2 0 0 ⎥
⎢ 0
𝚽s = ⎢
⎥=⎢
⎥
0
⋱
⋮ ⎥ ⎢⋮ 0 ⋱ ⋮ ⎥
⎢ ⋮
⎥ ⎢
⎥
⎢
⎣ 0
0
· · · ejkd sin 𝜃M ⎦ ⎣ 0 0 · · · zM ⎦
(8.27)
Finding 𝚽s requires knowing X1 and X2 .
The first subarray eigenvector matrix equals a matrix 𝚿 times the second
subarray eigenvector matrix.
Q[1 ∶ N − 1, 1 ∶ M] = 𝚿Q[2 ∶ N, 1 ∶ M]
(8.28)
Solving (8.28) for 𝚿 produces an estimate of 𝚽s . The eigenvalues of 𝚿 estimate
zm . The final step solves for the AOA estimates:
(
)
arg(𝜆Ψ
m)
−1
𝜃m = sin
(8.29)
kd
where 𝜆Ψ
m =eigenvalues of 𝚿.
Example
An eight-element uniform array with 𝜆/2 spacing has signals with amplitude
1.0 V/m incident at 𝜃 1 = 30∘ , 10∘ , 0∘ , and − 60∘ . Find the location of the signals
2
= 0.1 W∕m2 .
using the ESPRIT algorithm when 𝜎noise
Solution
The signal covariance matrix is given by (Figure 8.20)
–0.27273
–0.0849 + j0.4279
–0.1039 + j0.3159
0.1969 + j0.2940
–0.0326 + j0.3560
0.2697 + j0.2388
0.2826 + j0.2970
0.2232 – j0.1552
–0.43598
–0.1165 + j0.2158
0.2050 – j0.1770
–0.2418 – j0.3517
–0.4230 – j0.0715
–0.0611 + j0.2319
0.1199 – j0.2325
–0.1977 – j0.3811
0.36286
0.27961
0.0675 – j0.2691 0.4587 + j0.0905
–0.2907 + j0.3595 0.2856 – j0.1406
–0.0458 – j0.2728 0.0899 – j0.3054
0.0528 + j0.2742 –0.2870 + j0.1121
–0.3951 – j0.2117 –0.1195 + j0.3034
0.1924 + j0.2325 0.1291 + j0.4473
–0.3416 + j0.1200 0.0363 + j0.2851
Subarray 2
Subarray 1
8.5 Direction Finding Algorithms
Figure 8.20 Esprit covariance matrix.
Use (8.28) to compute 𝚿
⎡ 0.5960 − 0.2807i −0.4668 − 0.1233i 0.2495 + 0.5460i
⎢−0.0760 − 0.3009i 0.2589 − 0.7582i 0.1903 + 0.1133i
𝚿=⎢
−0.2309 − 0.5797i 0.1916 + 0.2367i −0.6670 + 0.3268i
⎢
⎣−0.1525 + 0.0351i −0.2582 − 0.1944i −0.0379 + 0.1709i
−0.0297 + 0.3218i⎤
−0.3489 − 0.4410i⎥
−0.2628 − 0.2593i⎥
⎥
0.7402 − 0.4018i⎦
The eigenvalues of 𝜓 are
[
]
𝜆Ψ = 0.92 + j0.41 −0.00 − j0.99 1.00 − j0.01 0.85 − j0.53
(8.30)
Substituting the eigenvalues into (8.29) yields an estimate of the AOAs.
[
]
𝜃 = −59.86∘ 0.21∘ 10.17∘ 30.08∘
m
8.5.7
Estimating and Finding Sources
Many DF algorithms need to know the number of signals incident on the array.
One approach to estimating the number of signals sets a threshold for noise
eigenvalues. Eigenvalues above the threshold belong to signals while the rest
belong to noise. One algorithm for estimating M is [16]
1. Estimate the covariance matrix from K time samples.
2. Find and sort the eigenvalues (𝜆1 > 𝜆2 > · · · > 𝜆N ).
3. Find M that minimizes
⎤
⎡
∑N
⎥
⎢
n=M+1 𝜆n
K(N − M) ln ⎢
) 1 ⎥ + f (M, N)
(∏
N−M ⎥
N
⎢ (N − M)
⎦
⎣
n=M+1 𝜆n
(8.31)
321
322
8 Direction Finding
where
{
f (M, N) =
M(2N − M)
0.5M(2N − M) ln N
Akaike’s information criterion
minimum description length
Problems
8.1
Derive (8.10).
8.2
Use a periodogram to demonstrate the effect of separation angle between
two sources using an eight-element uniform array with 𝜆/2 spacing when
𝜃 1 = − 30∘ , 10∘ , 20∘ and 𝜃 2 = 30∘ .
8.3
An eight-element uniform array with 𝜆/2 spacing has three signals
incident upon it: s1 (−60∘ ) = 1, s2 (0∘ ) = 2, and s3 (10∘ ) = 4. Find the Capon
spectrum.
8.4
An eight-element uniform array with 𝜆/2 spacing has three signals
incident upon it: s1 (−60∘ ) = 1, s2 (0∘ ) = 2, and s3 (10∘ ) = 4. Find the MEM
spectrum.
8.5
An eight-element uniform array with 𝜆/2 spacing has three signals
incident upon it: s1 (−60∘ ) = 1, s2 (0∘ ) = 2, and s3 (10∘ ) = 4. Find the MUSIC
spectrum.
8.6
An eight-element uniform array with 𝜆/2 spacing has three signals incident upon it: s1 (−60∘ ) = 1, s2 (0∘ ) = 2, and s3 (10∘ ) = 4. Find the location of
the signals using the root MUSIC algorithm.
8.7
An eight-element uniform array with 𝜆/2 spacing has three signals incident upon it: s1 (−60∘ ) = 1, s2 (0∘ ) = 2, and s3 (10∘ ) = 4. Estimate the incident
angles using ESPRIT.
References
1 Haupt, R.L. (2015). Timed Arrays Wideband and Time Varying Antenna
Arrays. Hoboken, NJ: Wiley.
2 Haupt, R.L. (2010). Antenna Arrays: A Computational Approach. Hoboken,
NJ: Wiley.
3 Frater, M.R. and Ryan, M. (2001). Electronic Warfare for the Digitized
Battlefield. Norwood, MA: Artech House.
References
4 Bayliss, E.T. (1968). Design of monopulse antenna difference patterns with
low sidelobes. The Bell System Technical Journal 47: 623–650.
5 Adcock, F. (1917). Improvement in means for determining the direction of a
distant source of electromagnetic radiation. British Patent 1304901919.
6 Baghdady, E.J. (1989). New developments in direction-of-arrival mea-
7
8
9
10
11
12
13
14
15
16
surement based on Adcock antenna clusters. In: Proceedings of the IEEE
Aerospace and Electronics Conference, 1873–1879. Dayton, OH, May 22–26:
IEEE.
Monzingo, R.A., Haupt, R.L., and Miller, T.W. (2011). Introduction to Adaptive Antennas, 2e. Scitech Publishing.
Capon, J. (1969). High-resolution frequency-wavenumber spectrum analysis.
Proceedings of the IEEE 57 (8): 1408–1418.
Pisarenko, V.F. (1973). The retrieval of harmonics from a covariance function. Geophysical Journal International 33 (3): 347–366.
Schmidt, R. (1986). Multiple emitter location and signal parameter estimation. IEEE Transactions on Antennas and Propagation 34 (3): 276–280.
Schmidt, R. and Franks, R. (1986). Multiple source DF signal processing: an
experimental system. IEEE Transactions on Antennas and Propagation 34
(3): 281–290.
Barabell, A. (1983). Improving the resolution performance of
eigenstructure-based direction-finding algorithms. In: ICASSP ’83. IEEE
International Conference on Acoustics, Speech, and Signal Processing, Boston,
Massachusetts, USA, 336–339.
Burg, J.P. (1972). The relationship between maximum entropy spectra and
maximum likelihood spectra. Geophysics 37 (2): 375–376.
Lacoss, R.T. (1971). Data adaptive spectral analysis methods. Geophysics 36
(4): 661–675.
Paulraj, A., Roy, R., and Kailath, T. (1986). A subspace rotation approach to
signal parameter estimation. Proceedings of the IEEE 74 (7): 1044–1046.
Godara, L.C. (2004). Smart Antennas. Boca Raton, FL: CRC Press.
323
325
9
Adaptive Arrays
Adaptive arrays modify their receive and/or transmit pattern characteristics
in response to the signals in the environment. The array evolves with time in
order to improve signal reception. A computer algorithm dynamically adjusts
the weights and switches in response to feedback, such as signal to noise ratio
(SNR). This chapter divides adaptive arrays into three categories: (i) adaptive
nulling, (ii) reconfigurable antennas, and (iii) beam-switching antennas. The
last two topics receive brief introductions at the end of the chapter. Adaptive
nulling occupies the vast majority of space.
9.1 The Need for Adaptive Nulling
As wireless users crowd the frequency spectrum, interference becomes more
common. When the sidelobe gain times the interference signal becomes large
enough to drop the SNR or bit error rate (BER) below an acceptable level, then
an adaptive array adjusts the element weights to improve performance. An
adaptive array maximizes the main beam gain in the direction of the desired
signal while minimizing the antenna pattern in the directions of the interfering
signals. Howells and Applebaum developed the first adaptive nulling antenna
for a radar [1, 2]. Their accomplishment went unrecognized until it was declassified a number of years later. The Howells–Applebaum algorithm must know
the direction of the desired signal in order avoid placing a null in that direction. Widrow developed a similar algorithm for communications systems [3].
His least mean square (LMS) algorithm serves as the basis for many adaptive
antennas. This algorithm needs to know the desired signal characteristics but
not its direction.
Consider the situation in Figure 9.1 in which a desired signal enters the
main beam (signal 1) and interfering signals 2 and 3 enter the sidelobes of
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
326
9 Adaptive Arrays
Figure 9.1 Three signals
incident on the quiescent
antenna pattern.
Signal 2
Signal 1
Signal 3
the quiescent pattern (pattern before adaptive algorithm places the nulls).
Assume that the desired signal has an amplitude of 1 V/m and the interfering
signals are 0.2 V while ignoring the noise. The desired signal received by the
antenna equals the desired signal power times the gain of the main beam.
Ps = (1)2 × 1 = 1 W
(9.1)
If the array factor sidelobe gains are −13 and −17 dB, then the received interference power is
PI = (0.2)2 × 10−13∕10 + (0.2)2 × 10−17∕10 = 0.0028 W
(9.2)
which leads to an signal to interference ratio (SIR) of 25.5 dB. In this case, the
interfering signals have a small impact on system performance.
Now, consider the situation in which signal 2 has an amplitude of 10 V/m
while the other two signals are 1 V/m. The desired received signal power is the
same, but the received interference power becomes
PI = (10)2 × 10−13∕10 + (1)2 × 10−17∕10 = 5.03 W
(9.3)
The SIR equals −7.02 dB which means the interference overwhelms the desired
signal.
Most communication systems require an SIR much larger than 0 dB, so the
antenna must mitigate the interference in order to maintain the link. A low sidelobe amplitude taper improves reception of the desired signal by decreasing
the received interference power relative to the received desired signal power.
For instance, reducing the first sidelobe by more than 20 dB reduces the interference power below that of the desired signal. Achieving such low sidelobes,
especially in small arrays, is nearly impossible and is very expensive. A static
low sidelobe amplitude taper battles interference from all angles outside the
main beam.
Adaptive nulling fights interference by dynamically changing the amplitude
and phase weights at the elements in order to place nulls in the antenna
pattern in the directions of the interfering signals. In a time varying signal
environment, the adaptive algorithm continuously updates the array element
weights in order to keep nulls pointing at the undesired signals as shown in
Figure 9.2.
9.2 Beam Cancellation
Figure 9.2 An adaptive array
starts with the quiescent pattern
in Figure 9.1 and places nulls in
the directions of the two
interfering signals while keeping
the main beam pointing at the
desired signal.
Signal 2
Signal 1
Signal 3
9.2 Beam Cancellation
Beam cancellation subtracts a cancellation beam (an array factor) from the quiescent array factor in order to place a null at a desired angle. When the array
encounters M interfering signals coming from 𝜃 m for 1 ≤ m ≤ M, then adaptation requires steering, weighting, then subtracting M cancellation beams from
the quiescent pattern [4]
N
∑
n=1
wn ejk(n−1)d sin 𝜃 =
N
∑
an ejk(n−1)d sin 𝜃
n=1
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟
quiescent array factor
−
M
∑
m=1
𝛾m
N
∑
bn ejk(n−1)d sin 𝜃 e−jkd(n−1)d sin 𝜃m
(9.4)
n=1
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
cancellation beams
The phase shift −kd(n − 1)d sin 𝜃 m steers the cancellation beam to the interference location 𝜃 m . Next, 𝛾 m weights beam m until it has the same gain as
the quiescent sidelobe level at 𝜃 m . M cancellation beams place M nulls when
subtracted from the quiescent pattern if the beams are orthogonal. Beam
orthogonality means that M − 1 beams have nulls in the direction of the peak
of the Mth beam. The adapted weights in (9.4) can be found using
M
bn ∑
wn = 1 −
𝛾 e−jk(n−1)d sin 𝜃m
an m=1 m
(9.5)
The cancellation beam amplitude taper, bn , controls the sidelobes and
beamwidth of the cancellation beams. Usually, the cancellation beam weights
either equal the amplitude taper of the quiescent array (bn = an ) or are uniform
(bn = 1) [5].
Example
Plot the quiescent and adapted array factors as well as the cancellation beams
of an eight-element array having a 20-dB Chebyshev amplitude taper with 𝜆/2
spacing when unwanted signals are incident at 𝜃 = 40∘ and − 25∘ .
327
9 Adaptive Arrays
Adapted
Quiescent
Cancellation beam
0
Directivity (dB)
328
–10
–20
–30
–40
–50
–60
–90
0
θ (deg)
–45
45
90
Figure 9.3 Nulls are created in the Chebyshev quiescent pattern by subtracting two
cancellation beams at 𝜃 = 40∘ and − 25∘ .
Solution
The Chebyshev weights are found from the synthesis approach in Chapter 4:
w = [0.5799 0.6603 0.8751 1.0000 1.0000 0.8751 0.6603 0.5799]. The height
of
[ the sidelobes in the directions of
] the interfering signals are given by 𝛾 =
0.0054 − j0.0825 0.0660 + j0.0659 . The adapted weights calculated from
(9.5) when an = bn are [0.9465 + j0.0132 1.0289 + j0.0998 1.6087 − j0.1480
1.7607 − j0.2905 1.7607 + j0.2905 1.6087 + j0.1480 1.0289 − j0.0998 0.9465 −
j0.0132]. Figure 9.3 shows the quiescent pattern superimposed on the adapted
pattern and two cancellation beams.
9.3 Optimum Weights
The error signal equals the magnitude of the difference between the desired
signal (s1 ) and the array output.
𝜀 = |s1 − wT (As + n)|
where
(9.6)
[
]T
s = s1 s2 · · · sM = signals
]T
[
w = w1 w2 · · · wN = element weights
⎡1
⎢ejkd sin 𝜃1
A=⎢
⋮
⎢ jk(N−1)d sin 𝜃1
⎣e
1
ejkd sin 𝜃2
⋮
ejk(N−1)d sin 𝜃2
···
···
⋱
···
1
⎤
⎥
ejkd sin 𝜃M
⎥
⋮
jk(N−1)d sin 𝜃M ⎥
e
⎦
= array steering matrix
]T
[
n = 𝜈1 𝜈2 · · · 𝜈N = element noise
9.4 Least Mean Square (LMS) Algorithm
The mean square error of (9.6) equals the expected value of the magnitude of
the error squared
E{𝜀2 } = E{|s1 |2 } + w† Cw − 2w† E{s1 (As + n)}
(9.7)
where the covariance matrix is given by
C = E{(As + n)(As + n)† }
(9.8)
Assuming that 𝜀 varies slowly with time, then taking the gradient of (9.7) with
respect to the element weights and setting it equal to zero leads to the minimum
of the mean square error
∇Tw E{𝜀2 } = 2Cwopt − 2E{s1 (As + n)}T = 0
(9.9)
Solving (9.9) for w produces the Wiener–Hopf solution [6]:
wopt = C−1 E{s1 (As + n)}T
(9.10)
Most adaptive algorithms strive to find weights close to wopt .
9.4 Least Mean Square (LMS) Algorithm
In an adaptive algorithm, the weight vector updates by an incremental amount
each time step until the interference disappears. The new weights at time sample n + 1 are the old weights at time sample n plus an incremental update, Δw[n]
w[n + 1] = w[n] + Δw[n]
(9.11)
Appropriate choices for Δw[n] push the weights closer to wopt . The gradient of
the square of the error with respect to the weights points toward the minimum
of 𝜀2 , so it directs the weight vector toward wopt .
w[n + 1] = w[n] − 𝜇0 ∇Tw 𝜀2
(9.12)
Using a trick to rewrite the gradient in (9.12) produces
𝜕𝜀2 𝜕𝜀
= w[n] − 2𝜇0 𝜀(As[n] + n[n])
𝜕𝜀 𝜕w
Finally, the LMS algorithm is given by [7]
w[n + 1] = w[n] − 𝜇0
(9.13)
w[n + 1] = w[n] + 𝜇{(As[n] + n[n])[s1 [n] − w† [n](As[n] + n[n])]}
(9.14)
where 𝜇 is a step size that includes the constants resulting from taking the
derivative. A large 𝜇 causes the algorithm to overshoot the optimum weights,
while a small 𝜇 causes very slow convergence.
329
9 Adaptive Arrays
The LMS algorithm remains stable when the step size stays within bounds
determined by the maximum eigenvalue (𝜆max ) of the covariance matrix [8]
0≤𝜇≤
2
(9.15)
𝜆max
Increasing 𝜆max /𝜆min slows the convergence speed, because (9.7) has a long, narrow valley that requires many iterations to find the minimum.
The LMS algorithm in (9.14) contains the desired signal which in practice
means an approximation for the desired signal, s1 (n). Approximations for s1 (n)
in (9.14) should [7]:
1. Be highly correlated with the desired signal and uncorrelated with the interference signals.
2. Have similar directional and spectral characteristics as those of the desired
signal.
Fortunately, these conditions are not difficult to meet.
Example
An eight-element array with a uniform amplitude taper and 𝜆/2 spacing has two
interference signals: a 2 V/m signal at 𝜃 = 61∘ and a 4 V/m signal at 𝜃 = − 21∘
while the desired signal is 1 V/m signal at 𝜃 = 0∘ . Use the LMS algorithm to
reduce the power received by the interfering signals while increasing the SNR.
Solution
First, form the covariance matrix then find the largest eigenvalue which is
30.6182. According to (9.15), 0 ≤ 𝜇 ≤ 0.0653. In order to have a slow, steady
convergence, 𝜇 is chosen to be 0.0001. Figure 9.4 shows a plot of the amplitude
and Figure 9.5 a plot of the phase of the adapted weights as a function of time.
They seem to settle at about 0.02 ms. Random noise causes the weights to
jitter. Figure 9.6 is the adapted pattern with the desired nulls superimposed on
the quiescent pattern. The output signal is compared to the transmitted signal
in Figure 9.7. As the weights settle down at 0.02 ms, the received signal looks
like the transmitted signal. Figure 9.8 has a plot of the signal to interference
Figure 9.4 LMS amplitude
weights as a function of time.
1
│Wn│
330
0.8
0.6
0.4
0
0.02
Time (ms)
0.04
9.4 Least Mean Square (LMS) Algorithm
Figure 9.5 LMS phase weights as
a function of time.
∠ Wn (deg)
40
20
0
–20
–40
0
0.02
0.04
Time (ms)
10
Directivity (dB)
Figure 9.6 The LMS adapted
pattern has nulls in the directions
of the interfering signals.
0
Adapted
Quiescent
–10
–20
–30
–50
0
50
Figure 9.7 The received signal
due to the LMS weights begins to
track the desired signal at about
0.02 ms.
Amplitude (V)
θ (deg)
5
0
–5
0
0.01
0.02
0.03
0.04
0.05
0.04
0.05
Time (ms)
100
SINR (dB)
Figure 9.8 Plot of SINR as a
function of time for the LMS
adaptive algorithm.
50
0
–50
0
0.01
0.02
0.03
Time (ms)
331
332
9 Adaptive Arrays
plus noise ratio (SINR) as a function of time. SINR starts at −1.37 dB and ends
at 28.1 dB. The SINR goes above 10 dB at 12.5 μs.
9.5 Sample Matrix Inversion Algorithm
In practice, the expected value needed to calculate the covariance matrix in
(9.10) is replaced by an average of a finite number of signal samples. There is
a tradeoff between the number of samples and the accuracy and convergence
speed of the algorithm. A single sample of the covariance matrix contains too
much noise to accurately represent the covariance matrix. Averaging decreases
the noncoherent noise while enhancing the coherent signals. Taking more samples requires time that slows down the adaptive algorithm.
̂ averages N s instantaneous samples of the
The sample covariance matrix, C,
covariance matrix
Ns
∑
̂= 1
(As[n] + n[n])(As[n] + n[n])†
C
Ns n=1
(9.16)
The sample matrix inversion (SMI) or direct matrix inversion (DMI) algorithm
consists of calculating the sample covariance matrix in (9.16) then substituting
into (9.17) to find the weights [6, 9]
̂ −1 {s1 [n](As[n] + n[n])}T
w=C
(9.17)
Example
An eight-element array with a uniform amplitude taper and 𝜆/2 spacing has two
interference signals: a 2 V/m signal at 𝜃 = 61∘ and a 4 V/m signal at 𝜃 = − 21∘
while the desired signal is 1 V/m signal at 𝜃 = 0∘ . Use the SMI algorithm to
reduce the power received by the interfering signals while increasing the SNR.
Solution
Figures 9.9 and 9.10 show plots of the amplitude and phase of the adapted
weights as a function of time. The amplitudes settle at about 0.02 ms, while
the phase settles at about 0.003 ms. The random noise and random variations
in the interfering signals cause the weights to jitter. Figure 9.11 is the adapted
pattern with the desired nulls superimposed on the quiescent pattern. The output signal is compared to the transmitted signal in Figure 9.12. The received
signal looks like the transmitted signal at a very early time in the adaptation.
Figure 9.13 has a plot of the SINR as a function of time. SINR starts at 0.7 dB
and ends at 45.6 dB. The SINR goes above 10 dB at 0.6 μs.
9.5 Sample Matrix Inversion Algorithm
Figure 9.9 SMI amplitude weights
as a function of time.
│Wn│
1
0.5
0
Figure 9.10 SMI phase weights as a
function of time.
0
0.02
Time (ms)
∠ Wn
500
0
–500
0
0.02
Time (ms)
0.04
10
Directivity (dB)
Figure 9.11 The SMI adapted
pattern has nulls in the directions
of the interfering signals.
0.04
Adapted
Quiescent
0
–10
–20
–30
–50
0
50
Figure 9.12 The received signal due
to the SMI weights begins to track the
desired signal at about 0.02 ms.
Amplitude (V)
θ (deg)
5
0
–5
0
0.01
0.02
0.03
Time (ms)
0.04
0.05
333
9 Adaptive Arrays
Figure 9.13 Plot of SINR as a
function of time for the SMI adaptive
algorithm.
60
40
SINR (dB)
20
0
–20
0
0.01
0.02
0.03
0.04
0.05
Time (ms)
9.6 Adaptive Algorithms Based on Power
Minimization
The adaptive algorithms described so far require digital beamforming in order
to quantify the signal at each element and form the covariance matrix. Most
phased arrays, however, do not have a receiver or analog to digital convertor
(ADC)/digital to analog converter (DAC) at each element. Instead, beamforming looks like the array in Figure 9.14. There are weights at each element, but
the only signal available to an adaptive algorithm is the sum of all the weighted
signals from the elements. As a result, only the output signal provides feedback
to an adaptive algorithm that adjusts the weights. The total output power of the
array is given by
P = |wT As|2
(9.18)
The output power in (9.18) has the desired and interfering signals mixed
together with the noise.
The desired signal power cannot be separated from the interference power
in (9.18). Consequently, minimizing the total output power of the array minimizes the desired signal as well as the interference [10]. Power minimization
only works as an adaptive nulling algorithm if the desired signal enters the main
beam and the interference enters the sidelobes. With this assumption, small
Figure 9.14 Power minimization
adaptive array.
Elements
w1
w2
w3
Weights
wN
Feedback
334
Power
Receiver
Computer
9.6 Adaptive Algorithms Based on Power Minimization
Figure 9.15 Phase shifter with four of its eight
bits adaptive.
Input signal
MSB
180°
45°
23°
11°
Adaptive
Beam steering
90°
Phase
shifter
6°
3°
1°
LSB
Output signal
weight perturbations have a small impact on the main beam, so the desired
signal remains relatively unaffected. Those same small weight perturbations
place nulls in the sidelobes to reduce interference. Two approaches to constraining weights to small variations are
• Partial adaptive nulling: Only a subset of the elements are adaptive [11, 12].
• Weight constraints: Weight variations limited to small values within a specified range [13, 14].
Partial adaptive nulling has N a adaptive elements out of a total of N elements.
N a should be large enough to form nulls in the highest sidelobes but small
enough to minimize the main beam gain loss. Feedback in Figure 9.14 only
goes to the N a elements. The remaining N − N a elements have static amplitude
weights and possibly beam steering phase shifters.
Making all the element weights adaptive requires weight constraints that
avoid nulling the desired signal entering the main beam. A constraint limits
the adaptive weights to the least significant bits (LSBs) of the amplitude and
phase weights. For instance, if the phase shifters have eight bit controls for
beam steering as shown in Figure 9.15, then the adaptive algorithm only has
access to the four LSBs. Four bits limit the maximum adaptive phase shift to
about 21∘ which is small enough to minimize the main beam loss but large
enough to place nulls in the sidelobes. All the bits steer the beam but the
adaptive algorithm only has access to the four LSBs create nulls.
9.6.1
Random Search Algorithms
Random, stochastic, or Monte Carlo search adaptive algorithms enhance the
desired signal and suppress the interfering signals by searching for the appropriate array weight values [7]. For the most part, random search algorithms
335
9 Adaptive Arrays
have slow convergence. They require minimal computations and hardware,
are insensitive to discontinuities or discrete variables, and find a minimum for
a multimodal cost function. Random search algorithms do not require digital
beamforming arrays which makes their implementation much easier and
cheaper. An adaptive array has an infinite number of weight values that reduce
the interference entering the sidelobes, so a random search algorithm has
good odds for successfully reducing the impact of interfering signals. Artificial
intelligence and machine learning offer guided random search algorithms that
control adaptive arrays [13].
Example
An eight-element uniform array has two adaptive elements: 1 and 8. Let the
amplitude of these elements lie between 0 and 1, while the phase varies between
0∘ and 360∘ . Find the lowest output power and the best SINR with 1000 random
guesses. Assume that the array has a uniform amplitude taper and 𝜆/2 spacing.
There are two interference signals: a 2 V/m signal at 𝜃 = 61∘ and a 4 V/m signal
at 𝜃 = − 21∘ while the desired signal is 1 V/m signal at 𝜃 = 0∘ .
Solution
Starting with a uniform array at trial 1, 999 random guesses are then made
for the two element weights. The resulting relative output power is shown in
Figure 9.16. The lowest output power of 18.0 dB is found at guess 235 compared
to the 24.8 dB output power for the uniform array. The weights associated with
this minimum relative output power, [0.6482 ∠ 79.5∘ , 1.0, 1.0, 1.0, 1.0, 1.0, 1.0,
0.7936 ∠ − 40.5∘ ], produce the adapted array factor shown in Figure 9.17. The
directivity decreases from 9.0 dB for the uniform array to 6.9 dB for the adapted
pattern. The SINR associated with the 1000 weight trials is shown in Figure 9.18.
The best SINR of 12.3 dB occurs at trial 235 compared to −1.4 dB for the uniform array. Only three guesses resulted in an SINR greater than 10 dB.
26
Output power (dB)
336
24
22
20
18
16
235
0
200
400
600
Random trial
800
Figure 9.16 Output power for 1000 random guesses of weights.
1000
9.6 Adaptive Algorithms Based on Power Minimization
Directivity (dB)
10
Adapted
Quiescent
0
–10
–20
–80
–60
–40
–20
0
20
40
60
80
θ (deg)
Figure 9.17 Adapted array factor superimposed on the quiescent (uniform) array factor.
15
235
SINR (dB)
10
5
0
–5
–10
0
200
600
400
Random trial
800
1000
Figure 9.18 SINR for 1000 random guesses of weights.
Example
An eight-element uniform array has all the elements adaptive. Weight variation are limited to: 0.8 ≤ |wn | ≤ 1.0 and 0.0 ≤ ∠ wn ≤ 72∘ . Find the lowest output
power and the best SINR with 1000 random guesses. Assume that the array has
a uniform amplitude taper and 𝜆/2 spacing. There are two interference signals:
a 2 V/m signal at 𝜃 = 61∘ and a 4 V/m signal at 𝜃 = − 21∘ while the desired signal
is 1 V/m signal at 𝜃 = 0∘ .
Solution
Starting with a uniform array at trial 1, 999 random guesses are then made
for the two element weights. The resulting relative output power is shown in
Figure 9.19. The lowest output power of 18.6 dB is found at guess 264 compared
to the 24.8 dB output power for the uniform array. The weights associated with
this minimum relative output power, [0.863 ∠ 71.8∘ , 0.908 ∠ 21.5∘ , 0.838 ∠ 4.4∘ ,
0.950 ∠ 41.2∘ , 0.858 ∠ 51.3∘ , 0.904 ∠ 33.9∘ , 0.944 ∠ 45.5∘ , 0.840 ∠ 1.4∘ ], produce
the adapted array factor shown in Figure 9.20. The directivity decreases from
9.0 dB for the uniform array to 6.7 dB for the adapted pattern. The SINR
337
9 Adaptive Arrays
Output power (dB)
28
26
24
22
20
264
18
0
200
400
600
Random trial
800
1000
Figure 9.19 Output power for 1000 random guesses of weights.
10
Directivity (dB)
338
Adapted
Quiescent
0
–10
–20
–80
–60
–40
–20
0
20
40
60
80
θ (deg)
Figure 9.20 Adapted array factor superimposed on the quiescent (uniform) array factor.
associated with the 1000 weight trials is shown in Figure 9.21. The best SINR
of 10.6 dB occurs at trial 264 compared to −1.4 dB for the uniform array. Only
one guess resulted in an SINR greater than 10 dB.
9.6.2
Output Power Minimization Algorithms
Guessing at weights and picking the values that yield the lowest output power
is a slow and inefficient way to do adaptive nulling. A better approach minimizes the calculated or measured output power of an array using numerical
minimization algorithms. This approach has been successfully verified through
experimental measurements as well as simulations [15, 16].
Example
An eight-element array has two adaptive elements: 1 and 8. Let the amplitude
of these elements lie between 0 and 1, while the phase varies between 0∘ and
360∘ . Use the Nelder–Mead downhill simplex algorithm [17] to minimize the
total output power.
9.6 Adaptive Algorithms Based on Power Minimization
15
264
SINR (dB)
10
5
0
–5
–10
0
200
400
600
Random trial
800
1000
Figure 9.21 SINR for 1000 random guesses of weights.
Solution
The resulting relative output power vs. iteration is shown in Figure 9.22. The
lowest output power of 16.95 dB is found after 187 iterations and 325 function
evaluations compared to the 20.71 dB output power for the uniform array. The
weights associated with this minimum relative output power, [0.5819 ∠ 67.6∘ ,
1 ∠ 0∘ , 1 ∠ 0∘ , 1 ∠ 0∘ , 1 ∠ 0∘ , 1 ∠ 0∘ , 1 ∠ 0∘ , 0.573 ∠ − 65.4∘ ], produce the adapted
array factor shown in Figure 9.23. The directivity decreases from 9.03 for the
uniform array to 6.11 dB for the adapted pattern. The SINR at the end of the
optimization is 20.92 dB compared to −1.37 dB for the uniform array. This is
faster and has a higher SINR than random guessing.
MATLAB commands:
options = optimset('Display','iter','PlotFcns',
@optimplotfval);
[y,fval,exitflag,output] =fminsearch('partialadapf',
rand(1,4),options)
The array output power is calculated in partialadapf.m.
Output power (dB)
24
22
20
18
16
0
50
100
Iteration
150
Figure 9.22 Convergence of the Nelder–Mead algorithm.
200
339
9 Adaptive Arrays
10
Directivity (dB)
340
Adapted
Quiescent
0
–10
–20
–80
–60
–40
–20
0
20
40
60
80
θ (deg)
Figure 9.23 Adapted array factor calculated from the optimized weights superimposed on
the quiescent (uniform) array factor.
Optimization algorithms are classified as either global or local searches.
Hybrid optimization combines the two by first starting a global search that
hands over its best result to a fast local optimizing algorithm. Global searches,
such as genetic algorithm or particle swarm optimization guide random
searches based on biological or physical processes in nature. These algorithms
have been successfully applied to experimental adaptive arrays [16].
9.6.3
Beam Switching
Some array beamforming networks host multiple overlapping beams that
point in slightly different directions. Figure 9.24 shows a diagram where one
of four beams is selected by closing the appropriate switch. Two examples
of multi-beam beamforming networks are the Rotman lens [18] and Butler
matrix [19]. Each beam port receives a signal from all the elements. Path
lengths differences from the elements to the beam ports cause a beam steering
phase shift that points the beams in different directions. The beams overlap
and cover the desired area.
9.6.4
Reconfigurable Antennas
Configuring an antenna means designing its shape, material properties, feed
location, etc., so that it radiates at a desired frequency and polarization. If
Figure 9.24 Beam switching with
a multi-beam antenna.
Beam 1
4
Beam 2
Beam 3
Beam 4
3
Array
beamforming
network
2
1
9.6 Adaptive Algorithms Based on Power Minimization
the operating characteristics of the antenna change, then the antenna needs
reconfigured to satisfy the new specifications. Reconfigurable antennas modify
their performance characteristics by changing the current flow on an antenna,
using mechanically movable parts, phase shifters, attenuators, diodes, tunable
materials, or active materials [20].
Radio frequency (RF) switches in an antenna cause currents to flow or not
flow along paths in order to change the antenna’s radiation properties, as well
as its impedance. Figure 9.25 shows a patch with switches that short slots when
closed in order to switch between polarizations.
Reconfigurable antenna arrays modify array properties to control the
antenna pattern. Figure 9.26 shows a hemispherical array for communicating
with satellites. This hemispherical array follows satellites anywhere in the
sky. Each triangular subarray is a planar array of antenna elements. Adjacent
triangles combine to form larger arrays, so the aperture size adjusts the amount
of gain need to communicate with a satellite. The active aperture moves across
the dome in order to follow a satellite across the sky.
Figure 9.25 A reconfigurable
slotted-patch antenna. Source:
Haupt and Lanagan [20].
Reproduced with permission of
IEEE.
Substrate
Feed
Patch
Switches
Slots
Side
Figure 9.26 Phased array on a
hemispherical surface for satellite
communications. Source: Haupt and
Lanagan [20]. Reproduced with
permission of IEEE.
Top
Expanded
Original
Moved
Elements
Subarray
341
342
9 Adaptive Arrays
Problems
1
Plot the quiescent and adapted array factors as well as the uniform cancellation beams of an eight-element uniform array with 𝜆/2 spacing when
unwanted signals are incident at 𝜃 = 61∘ and − 20∘ .
2
Plot the quiescent and adapted array factors as well as the 20 dB Chebyshev cancellation beams of an eight-element uniform array with 𝜆/2 spacing
when unwanted signals are incident at 𝜃 = 61∘ and − 20∘ .
3
Plot the adapted pattern resulting from placing a null at 𝜃 = 16.25∘ in the
array factor of a 32-element uniform array with 𝜆/2 element spacing using
random search. Four different configurations of eight adaptive elements are
considered:
(a) 1, 2, 3, 4, 29, 30, 31, 32
(b) 13, 14, 15, 16, 17, 18, 19, 20
(c) 1, 5, 9, 13, 17, 21, 25, 29.
4
Plot the adapted pattern resulting from placing a null at 𝜃 = 16.25∘ in the
array factor of a 32-element uniform array with 𝜆/2 element spacing using
random search. Four different adaptive weight limits assuming every element in the array is adaptive:
(a) 0.8 ≤ an ≤ 1.0, 0 ≤ 𝛿 n ≤ 72∘
(b) 0.8 ≤ an ≤ 1.0, 𝛿 n = 0∘
(c) an = 1.0, 0 ≤ 𝛿 n ≤ 72∘
(d) 0.5 ≤ an ≤ 1.0, 0 ≤ 𝛿 n ≤ 180∘ .
5
An eight-element uniform array with 𝜆/2 spacing has the desired signal incident at 0∘ and two interference signals incident at −21∘ and 61∘ . Use the
LMS algorithm to place nulls in the antenna pattern. Assume 𝜎 noise = 0.01.
In MATLAB, represent the signal by cos(2𝜋(1 : K)/K) exp(j rand) and the
interference by sign(randn(1, K)).
6
An eight-element uniform array with 𝜆/2 spacing has the desired signal incident at 0∘ and two interference signals incident at −21∘ and 61∘ . Use the
SMI algorithm to place nulls in the antenna pattern. Assume 𝜎 noise = 0.01.
In MATLAB, represent the signal by cos(2𝜋(1 : K)/K) exp(j rand) and the
interference by sign(randn(1, K)).
References
1 Howells, P. (1976). Explorations in fixed and adaptive resolution at GE and
SURC. IEEE Transactions on Antennas and Propagation 24 (5): 575–584.
References
2 Applebaum, S. (1976). Adaptive arrays. IEEE Transactions on Antennas and
Propagation 24 (5): 585–598.
3 Widrow, B., Mantey, P.E., Griffiths, L.J. et al. (1967). Adaptive antenna
systems. Proceedings of the IEEE 55 (12): 2143–2159.
4 Haupt, R.L. (2010). Antenna Arrays: A Computational Approach. Hoboken,
NJ: Wiley.
5 Steyskal, H., Shore, R.A., and Haupt, R.L. (1986). Methods for null control
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
and their effects on the radiation pattern. IEEE AP-S Transactions AP-34
(3): 163–166.
Compton, R.T. (1987). Adaptive Antennas: Concepts and Performance.
Philadelphia, PA: Prentice-Hall.
Monzingo, R.A., Haupt, R.L., and Miller, T.W. (2011). Introduction to Adaptive Antennas, 2e. Scitech Publishing.
Gross, F.B. (2005). Smart Antennas for Wireless Communications: With
MATLAB. New York: McGraw-Hill.
Gupta, I. (1986). SMI adaptive antenna arrays for weak interfering signals.
IEEE Transactions on Antennas and Propagation 34 (10): 1237–1242.
Haupt, R.L. (2015). Timed Arrays Wideband and Time Varying Antenna
Arrays. Hoboken, NJ: Wiley.
Morgan, D. (1978). Partially adaptive array techniques. IEEE Transactions
on Antennas and Propagation 26 (6): 823–833.
Haupt, R.L. and Shore, R.A. (1984). Experimental partially adaptive nulling
in a low sidelobe phased array. In: Antennas and Propagation Society International Symposium, 823–826. Boston, MA: IEEE.
Haupt, R.L. (1997). Phase-only adaptive nulling with a genetic algorithm.
IEEE Transactions on Antennas and Propagation 45 (6): 1009–1015.
Haupt, R.L. (2010). Adaptive nulling with weight constraints. Progress In
Electromagnetics Research B 26: 23–38.
Haupt, R.L. and Werner, D.H. (2007). Genetic algorithms in electromagnetics. Hoboken, NJ: IEEE Press: Wiley-Interscience.
Haupt, R.L. and Southall, H. Experimental adaptive cylindrical array.
Microwave Journal 3: 291–296.
Press, W.H., and Numerical Recipes Software (Firm) (1994). Numerical
Recipes in FORTRAN. Cambridge University Press.
Rotman, W. and Turner, R. (1963). Wide-angle microwave lens for line
source applications. IEEE Transactions on Antennas and Propagation 11 (6):
623–632.
Butler, J. and Lowe, R. (1961). Beam-forming matrix simplifies design of
electronically scanned antennas. Electronic Design 9: 170–173.
Haupt, R.L. and Lanagan, M. (2013). Reconfigurable Antennas. IEEE Antennas and Propagation Magazine 55 (1): 49–61.
343
345
10
MIMO
Multiple input/multiple output (MIMO) systems rely on diversity and adaptive signal processing in the transmit and receive array antennas to dramatically
increase data rates and spectral efficiency [1, 2]. Diversity increases as the number of transmit and receive antenna elements increase. Both the transmit and
receive arrays adapt their weights in order to emphasize productive subchannels between the transmit and receive elements in order to increase the desired
signal reception.
Figure 10.1 shows four categories of wireless communication systems based
on the number of elements at the transmitter (N t ) and receiver (N r ) [3]. Most
communication systems have one antenna transmitting to a receive antenna
(N t = 1, N r = 1) or single input single output (SISO). The signal travels through
a channel that has an impulse response given by h11 . SISO works well in a time
invariant, high signal to noise ratio (SNR) channel. Multiple input single output
(MISO) communication systems have a transmit array antenna and only one
receive antenna (N t > 1, N r = 1). A subchannel from transmit element m to the
receive antenna impulse response of hm1 . Single input multiple output (SIMO)
systems have multiple antennas at the receiver but the transmitter only has one
antenna (N t = 1, N r > 1). The subchannel impulse response from the transmit antenna to receive element n is given by h1n . MIMO has an antenna array
at the transmitter and receiver (N t > 1, N r > 1). It has a subchannel impulse
responses, hmn , between from each of the N t transmit antennas to each of the
N r receive antennas.
This chapter introduces the concept of MIMO and the critically important
channel matrix. The channel matrix leads to finding the receive and transmit
element weights that increase the channel capacity.
10.1 Types of MIMO
A MIMO communication system uses either spatial diversity or spatial
multiplexing techniques to increase data transfer [4]. A transmitter with one
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
10 MIMO
h11
h11
1
1
h21
2
hNt1
1
…
1
Nt
SISO
h11
h12
1
1
h1N
2
2
h11
h12
h21
h22
2
…
r
Nr
Nt
1
hNt 2
hNt1
h2Nr
hN N
…
1
MISO
…
346
h1Nr
Nr
r r
SIMO
MIMO
Figure 10.1 Possible antenna combinations at the transmitter and receiver.
antenna (Figure 10.2a) sends one stream of data from one location. Spatial
diversity at the transmitter means that each array transmit element sends the
same data signal to the receiver in order to increase the diversity of paths.
Each path encounters different fading, so more paths increase the probability
that one of them successfully delivers the signal. Figure 10.2b shows each of
the four transmitting antennas sending the binary string “1 0 0 1”. In this case,
the antenna in Figure 10.2a replicates four times to form the transmit array.
Placing transmit array elements far apart increases the odds that paths from
one transmit element to the receiver are independent. The diversity gain for
this system is N t = 4, but the data rate is same as the single transmit antenna.
Spatial-multiplexing at the transmitter assigns one bit in “1 0 0 1” to each of
the four transmit antennas as shown in Figure 10.2c. In this case, a different
data stream travels over each path or subchannel from a transmit element to a
receive element. This approach increases the data rate by four but the diversity
gain is zero. MIMO takes advantage of the gains provided by spatial diversity
and spatial multiplexing.
10.1 Types of MIMO
Figure 10.2 Transmitting
a data stream. (a) Single
transmitter, (b) spatial
diversity, and (c) spatial
multiplexing.
1001
1001
1001
(a)
0
1
h12
h21
2
2
h22
20
30
10
Time sample
h11
40
Amplitude (dB)
Amplitude (dB)
h11
50
0
0
1001
0
1001
1
(c)
−10
h12
0
10
20
30
Time sample
40
50
0
h21
0
10
20
30
Time sample
40
50
Amplitude (dB)
Amplitude (dB)
1001 1001
0
1
−10
1
(b)
0
−10
1001
−10
h22
0
10
20
30
Time sample
40
50
Figure 10.3 Channel attenuation in a 2 × 2 MIMO system as a function of time.
As an example, consider a MIMO system with an array of two elements on
transmit and two elements on receive. Figure 10.3 has plots of the signal levels at
the receive elements for the four paths in a 2 × 2 MIMO system over time. Each
channel experiences fades at different times due to the different signal paths
and a time-varying environment. A SISO system over h11 encounters strong
signal fades at time samples 6 and 35 in which the receiver may not detect the
signal. This 2 × 2 MIMO system has three other paths in which signal fades do
not occur at time samples 6 and 35, so the receiver ignores the signal from the
bad path and combines the signals from the good paths to optimize the signal
reception. The more the transmit and receive antennas in the system (more
diversity), the higher the probability that one of the N t × N r subchannels results
in a strong signal at the receiver.
If the signal transmitted from element m, stm (t), travels through a subchannel
with impulse response, hmn (t), then the signal arriving at receive element n is
given by
srn (t) = stm (t)∗ hmn (t)
(10.1)
347
348
10 MIMO
where “*” is convolution. A MIMO system uses the impulse response from each
transmit element to each receive element to increase channel capacity. Channel sounding finds the subchannel impulse responses by sending a broadband
signal with a flat frequency spectrum from each transmit element to each of the
receive element [5]. The broadband signal approximates an impulse function in
the time domain, so the output at each receive element looks like the impulse
response. Since the receiver knows the transmitted signal as well as the received
signal, it uses deconvolution to find hmn (t) [6]. Taking the Fourier transform of
(10.1) and solving for the channel transfer function produces
H( f ) =
Sr ( f )
St ( f )
(10.2)
The inverse Fourier transform of (10.2) results in the channel impulse
response. In practice, the receiver performs the deconvolution using the
z-transform. After deconvolving all the subchannel impulse responses go into
a N r × N t channel matrix.
A simplified MIMO system model helps illustrate how to build the channel
matrix. Assume that the channel does not change with time, and the signals are
single frequency carriers with no modulation. The first transmit antenna sends
a calibration signal to the three receive antennas. The received signals at the
elements go into the three rows of column 1 in the channel matrix. Next, transmit element 2 sends a signal so that the received signal goes into the second
column of the channel matrix. This process continues for transmit element 3.
In this way, each transmit array element sends a signal that each receive array
element records in a column of the channel matrix, H.
Consider a narrow band channel matrix measured at the receiver given by
⎡0.0756 + j0.0848 0.0185 + j0.0195 0.2522 + j0.0651⎤
⎢
⎥
H = ⎢0.0223 + j0.0032 0.0061 + j0.0112 0.1763 + j0.0230⎥
⎢0.0430 − j0.0131 0.0103 − j0.0010 0.0585 + j0.0266⎥
⎣
⎦
(10.3)
The measured values in column 2 are about a magnitude lower than those in
columns 1 and 3, so turning off transmit antenna 2 and distributing its power
to transmit antennas 1 and 3 increases the signal power delivered to the receive
array elements. This simple procedure is a fundamental MIMO concept.
MIMO commonly uses orthogonal frequency division multiplexing (OFDM)
to split the high-speed serial data into lower-speed serial data signals for each
transmit element [7]. The longer symbol periods reduce multipath time delays.
As mentioned in Chapter 3, when the subcarrier spacing equals the reciprocal
of the symbol period of the data signals, they are orthogonal. The resulting sinc
function frequency spectra have their first nulls at the subcarrier frequencies
on the adjacent channels.
10.2 The Channel Matrix
10.2 The Channel Matrix
The last section introduced a simple channel matrix example. This section starts
by deriving H from basic propagation principles. A MIMO system has N t transmit antennas sending signals to N r receive antennas. If an isotropic point source
transmits the signal stm to a point source at a receiver rmn0 away in free space,
then the received signal, srn , is given by
(
)
stm t − 𝜏mnp
srn (t) =
(10.4)
√
2 𝜋rmnp
The signal takes 𝜏 mnp to travel along path p of length rmnp from transmit element
m to receive element n. Path p = 0 is line of sight (LOS). A SISO system with
no multipath has n = 1, m = 1, and p = 0. The rmn0 path does not exist for a
Rayleigh channel.
In a channel with multipath and moving transmitter and/or receiver, the signal arriving at receive array element n is [8]
srn =
Nt
∑
n=1
=
Nt
∑
∞
∫−∞
hmn (𝜏, t)stm (t − 𝜏)d𝜏
hmn (t)∗ stm (t), m = 1, 2, … , Nr
(10.5)
n=1
The signal transmitted by each antenna travels a different path to get to
the receiver. Thus, each subchannel (e.g. from transmit antenna m to receive
antenna n) has an impulse response, hmn (𝜏, t). The variable 𝜏 represents the
time delay due to the different lengths of the multipath, and the t represents
the time-changing channel (Doppler). Putting (10.5) in matrix form results in
sr (t) = H(t)∗ st (t)
(10.6)
In a flat fading channel this equation reduces to
sr (t) = H(t)st (t)
(10.7)
which greatly simplifies the math. Stationary transmit and receive antennas
result in a static channel (t → 0).
sr (t) = Hst (t)
(10.8)
A narrow band subchannel has an inpulse response that is a complex constant.
The signal transmitted from element m arrives at receive array element n
via LOS and N p (m, n) single bounce multipath signals that have path lengths
rmnp and time of arrivals of 𝜏 mnp and reflection coefficients Γp (m, n). The
signal in a subchannel decreases in amplitude with each reflection. After many
reflections, the signal amplitude becomes very small and can be ignored.
349
350
10 MIMO
Computer programs based on shooting and bouncing rays (SBR) (Chapter 5)
eliminate all rays that fall below an amplitude threshold in order to make the
computations in complex models reasonable.
srn (t) =
Np (m,n)
∑ stm (t − 𝜏mnp )Γp (m, n)
stm (t − 𝜏mn0 )
+
√
√
2 𝜋rmn0
2 𝜋rmnp
p=1
(10.9)
The reflection coefficients depend on the angles of incidence and the electrical properties of the surfaces. As a result, H randomly changes with time
when the transmitter and/or receiver moves. A carrier signal transmitted into
a flat fading channel has an equivalency between time delay and phase shift by
substituting 𝜉 = krmnp = 2𝜋f𝜏
Np (m,n)
⎛ −jkrmn0
∑ Γp (m, n)e−jkrmnp ⎞
e
⎟ s (t)
srm (t) = ⎜ √
+
√
⎟ tn
⎜ 2 𝜋r
2
𝜋r
p=1
mn0
mnp
⎠
⎝
(10.10)
with the channel characterized by
Np (m,n)
∑ Γp (m, n)e−jkrmnp
e−jkrmn0
hmn = √
+
√
2 𝜋rmn0
2 𝜋rmnp
p=1
(10.11)
Figure 10.4 shows a MIMO system that has LOS paths from each transmit
element to each receive element as well as a one bounce multipath signal.
The relationship between the transmitted signals and the received signals in a
multipath environment becomes very complex but greatly simplifies for narrow
band signals:
sr = Hst + n
1
(10.12)
1
Nt
Nr
Figure 10.4 The LOS paths are
dashed lines while the one
bounce multipaths are solid
lines.
10.2 The Channel Matrix
where
N = N t × 1 noise vector
H = N r × N t channel matrix
st = N t × 1 transmit signal vector
sr = N r × 1 transmit signal vector.
Figure 10.1 labels subchannels between the transmit array elements and the
receive array elements with the appropriate subchannel impulse responses.
⎡ h11 h12 · · · h1Nt ⎤
⎢
⎥
⋮ ⎥
⎢ h21 h22
H=⎢
⎥
⋱
⎢
⎥
⎢
⎥
hNr Nt ⎦
⎣hNr 1 · · ·
(10.13)
Slow time variations (e.g. stationary transmitter and receiver) produce an H
with elements that are complex constants.
Chapter 8 introduced the signal covariance matrix for direction finding which
was also used for adaptive nulling in Chapter 9. Since MIMO has adaptive transmit and receive arrays, there are two signal covariance matrices of interest: one
N t × N t matrix for transmit (Cst ) and one N r × N r matrix for receive (Csr )
Csr = E[s†r sr ]
(10.14)
Cst = E[s†t st ]
(10.15)
The average total transmitted power is greater than or equal to the sum of the
powers radiated by all the elements:
Pt ≥ tr(Cst )
(10.16)
where tr(⋅) is the trace of a matrix or the sum of the elements along the main
diagonal. For a SISO communications system, the SNR at the transmitter is
defined by
SNRt =
Pt
2
𝜎noise
(10.17)
and at the receiver by
SNRr =
Pt |h11 |2
2
𝜎noise
(10.18)
Experimental measurements or a computer model generates the channel
matrix [9]. A switched array design has a single transmitter and a single
receiver that measures H by sequentially connecting all combinations of
transmit and receive array elements using high-speed switches. Switching
351
352
10 MIMO
times from 2 to 100 ms enable the measurement of all antenna pairs before the
channel appreciably changes for most environments. Virtual arrays displace or
rotate a single antenna to form an array over time. A complete channel matrix
measurement takes several seconds or minutes to perform, so the channel
must remain stationary over that time in order to be accurate. As a result,
virtual arrays work best for fixed indoor measurements with little motion.
10.3 Recovering the Transmitted Signal Using
the Channel Matrix
The simplified channel matrix in the previous section 10.2 highlights some of
the basic principles of multipath channels. In reality, propagation effects in the
channel impact the elements of the channel matrix in very complicated ways.
The channel state information (CSI) describes how a transmitted signal changes
due to the channel effects described in Chapter 5. Channel state information at
the transmitter (CSIT) requires the receiver to send its version of H back to
the transmitter. Channel state information at the receiver (CSIR) only happens
when the transmitter sends a test signal for calibration. MIMO has five different
cases of CSI [10]:
1.
2.
3.
4.
5.
CSIT and CSIR
No CSIT and CSIR
Statistical CSIT and CSIR
Noisy CSI
No CSIT and no CSIR.
This section covers the first two cases.
CSI is either instantaneous (long term) or statistical (short term). A slow
fading channel gives a system sufficient time to develop a reasonable estimate
of the channel matrix. In this case, a channel matrix estimate slowly changes,
so it does not need constant updating. On the other hand, a fast-fading
channel matrix requires updating before the channel changes again, so channel
statistics, such as the fading distribution, average channel gain, and spatial
correlation replace actual measurements.
10.3.1
CSIR and CSIT
In a system with CSIR and CSIT, the receiver recovers the transmitted data
by inverting the channel matrix (N t = N r ) and multiplying the received signal
vector (ignoring noise).
st = H−1 sr
(10.19)
10.3 Recovering the Transmitted Signal Using the Channel Matrix
which requires an invertible H. When N t ≠ N r , a least-squares solution approximates the desired signal:
ŝ t = (H† H)−1 H† sr
(10.20)
The channel matrix properties (e.g. condition number and rank) predict the
solution accuracy when solving (10.19) and (10.20).
Channel sounding estimates H by measuring the subchannel propagation
characteristics. A full rank H has linearly independent rows and columns due to
independent paths. A low matrix condition number indicates a full rank matrix
(all rows and columns are linearly independent). MIMO channels with minimal multipath or a large separation distance between the transmit and receive
antenna arrays have a nearly singular H [8]. A low-rank MIMO channel (has
many linearly dependent rows and columns) behaves like a SISO channel with
the same total power. A high multipath environment, on the other hand, has an
H with high rank.
MIMO excels in a high multipath environment with no LOS signal, because H
appears random and has a low condition number. MIMO does not work well in
the presence of strong LOS signals and no multipath. As an example, a MIMO
system in free space with no multipath reduces H to
−jkr
−jkr
−jkr
⎡ e 110 e 120 · · · e 1Nt 0 ⎤
⎢ r110
r120
r1Nt 0 ⎥
⎢
⎥
⎢ e−jkr210 e−jkr220
⎥
⎥
1 ⎢ r
r220
H = √ ⎢ 210
⎥
2 𝜋⎢
⋮
⋱
⋮ ⎥
⎢
⎥
⎢ e−jkrNr 10
e−jkrNr Nt 0 ⎥
⎢
⎥
···
⎣ rN 10
rNr Nt 0 ⎦
r
(10.21)
In this case, H is ill-conditioned and (10.21) has a high condition number. As
a result, the calculated st has numerical errors. Increasing the element spacing
in the transmit and receive arrays and/or adding multipath to the subchannels
decorrelates the matrix elements and decreases the matrix condition number.
The number of data streams (transmitted packets of data) supported is less
than or equal to the rank of H, where the rank of a matrix is the maximum number of linearly independent rows or columns. Singular values are the positive
square roots of the nonzero eigenvalues of H† H and indicate which transmit
subchannels deliver high-quality signals [11]. The singular value decomposition
(SVD) of H extracts the singular values by decomposing it into
H = UDV†
(10.22)
353
354
10 MIMO
⎡𝜆1 0 0 ⎤
⎥
⎢
D = ⎢ 0 ⋱ 0 ⎥ = Nr × Nt diagonal matrix
⎥
⎢
⎣ 0 0 𝜆Nt ⎦
𝜆m = singular value
U = N r × N r column orthonormal matrix
V = N t × N t column orthonormal matrix.
The V matrix weights the data at the transmitter, while the U matrix weights
the received signals. The singular values in D correspond to relative subchannel
weights.
Example
Show that the singular values of H are the positive square roots of the nonzero
eigenvalues of H† H when
⎡0.0756 + j0.0848 0.0185 + j0.0195 0.2522 + j0.0651⎤
⎢
⎥
H = ⎢0.0223 + j0.0032 0.0061 + j0.0112 0.1763 + j0.0230⎥
⎢
⎥
⎣0.0430 − j0.0131 0.0103 − j0.0010 0.0585 + j0.0266⎦
Solution
Create an m-file that has the matrix H then use the commands:
[U,S,V] = svd(H);
[E,D] = eig(H'*H);
[S sqrt(D)]
to get the output:
0.3407
0
0
0.3407
0
0
0
0.0623
0
0
0.0623
0
0
0
0.0059
0
0
0.0059
The SVD decomposition provides insight into MIMO performance as shown
in Figure 10.5. Substitute (10.22) into (10.12) to get
sr = UDV† st + N
Data
∼
s
t
(10.23)
Precoding
st =
Vs∼t
Channel
st
Shaping
sr = Hst + N
Transmitted
signal
sr
∼
s
r=
U†sr
Received
signal
Figure 10.5 Transmitter precoding and receiver shaping.
Data
∼
s
r
10.3 Recovering the Transmitted Signal Using the Channel Matrix
The transmitter precodes the data signals (̃st ) before sending them to the
transmit array elements:
(10.24)
st = Ṽst
Finally, shaping recovers the data (̃sr ) at the receiver:
s̃ r = U† sr
(10.25)
Transmit precoding requires that the transmitter knows V (CSIT), so a system
with CSIT and CSIR uses SVD.
Example
A MIMO system with two transmit elements and three receive elements has
the channel matrix below, find the SVD decomposition.
⎡0.8268 0.8558⎤
⎢
⎥
H = ⎢0.6825 0.7192⎥
⎢0.5895 0.3907⎥
⎣
⎦
Solution
The MATLAB command svd(H) decomposes H:
⎡− 0.7009
⎢
H = UDV = ⎢− 0.5838
⎢− 0.4098
⎣
[
− 0.7187
−0.2837 −0.6544⎤ ⎡1.6968
0 ⎤
⎥⎢
⎥
−0.2989 0.7549 ⎥ ⎢ 0
0.1417⎥
0.9111 0.0438 ⎥⎦ ⎢⎣ 0
0 ⎥⎦
]
0.6953
− 0.6953 −0.7187
There are two independent channels with gains of 1.6968 and 0.1417. The
channel with the largest singular value is the most reliable.
Alternatively, the output from the receive array can be written as a function
of the singular values. Begin by substituting (10.24) into (10.23) then that result
into (10.25) to get
s̃ r = U† UDV† Ṽst + U† N
(10.26)
which simplifies to [12]
̃
s̃ r = D̃st + N
(10.27)
̃ has the
If U and V are unitary matrices (i.e. U U = INr and V V = INt ), then N
same statistical properties as N. Since D is a diagonal matrix of singular values,
the data in a row of s̃ r are array weights (singular values) that multiply the data
in the corresponding row of s̃ t plus noise. When the matrix rank is less than N t ,
†
†
355
10 MIMO
N1
1
1
+
sr1
+
sr2
2
…
s∼tN
2
…
…
N2
st2
k
0
stNt
0
Nr
Nt
∼
sr1
…
st1
∼
NNk
+
srNk
srNr
…
s∼t1
…
0
0
Figure 10.6 A MIMO system transmits and decodes less than or equal to Nk data streams.
λ1
N1
∼
s
t1
Figure 10.7 Equivalent model for Figure 10.6.
∼
s
r1
…
356
λN
∼
stNk
k
NN
k
∼
srNk
then only the data streams s̃t1 to s̃tN k get transmitted. Figure 10.6 illustrates how
a MIMO system only transmits N k data streams. The receiver only decodes up
to N k data streams. Figure 10.7 simplifies Figure 10.6 to a model of (10.27).
A time-invariant MIMO system with CSIT and CSIR has the channel capacity: [13]
)]
[
(
1
†
bps
CCSIT-CSIR = E maxCst ;tr(Cst )≤P B log2 det INt + 2 HCst H
𝜎noise
(10.28)
where the maximization is over the N t × N t input covariance matrix Cst and
SNRt is the average SNR in the subchannels.
10.3.2
Waterfilling Algorithm
The waterfilling algorithm implements (10.26) by allocating more transmit
power to higher SNR subchannels and less power to low SNR subchannels
10.3 Recovering the Transmitted Signal Using the Channel Matrix
[7]. Its name comes from refilling water glasses until they all have an equal
amount of water. A glass containing water gets less water from the pitcher
than a glass that has no water. In MIMO waterfilling, the amount of transmit
power allocated to a subchannel is proportional to the SNR in that subchannel.
Just like the empty glass gets more water, a high SNR subchannel gets more
transmit power.
A MIMO subchannel has N k subchannels associated with the N k singular
values of the SVD. Waterfilling allocates power to the subchannels up to a level
of SNR0 according to
[(
) ]
1
1
Pn = max
−
,0
(10.29)
SNR0 SNRn
The value of SNR0 is chosen in order that
Nk
∑
(10.30)
Pn = Pt
n=1
and the subchannel SNRs are given by
SNRn =
𝜆2n
(10.31)
2
𝜎noise
Figure 10.8 diagrams the waterfilling process for a MIMO system with six
subchannels. Subchannels with SNRn ≤ SNR0 are not allocated any power, so
the transmitter allocates Pt to the four subchannels according to (10.29). When
2
is high, the optimum power distribution evenly allocates power across
Pt ∕𝜎noise
Figure 10.8 Example of
water filling with six
subchannels.
1
SNR0
P1
P2
P3
P4
1
SNR1
1
SNR2
1
SNR3
1
SNR4
1
2
3
4
Subchannel
P5 = 0 P6 = 0
1
SNR5
1
SNR6
5
6
357
358
10 MIMO
1
SNR0
P1
P2
P3
P4
1
SNR1
1
SNR2
1
SNR3
1
SNR4
1
2
4
3
Subchannel
(a)
P5 = 0 P6 = 0
1
SNR5
1
SNR6
5
6
P1
P2 = 0
P3 = 0 P4 = 0
P5 = 0 P6 = 0
1
SNR1
1
SNR2
1
SNR3
1
SNR5
1
SNR6
1
2
5
6
1
SNR4
3
4
Subchannel
(b)
Figure 10.9 Waterfilling
2
is high and
when Pt ∕𝜎noise
low. (a) High SNR and (b)
low SNR.
1
SNR0
all subchannels as shown in Figure 10.9a. The channel capacity for high SNR is
[14]
(
)
Pt
bps
(10.32)
Chi = Nk B log2
2
𝜎noise
2
At the other extreme, when Pt ∕𝜎noise
is low (Pt < 1/𝜆2 − 1/𝜆1 ), all the power
goes into the subchannel with the best SNR (Figure 10.9b). The channel capacity
10.3 Recovering the Transmitted Signal Using the Channel Matrix
for low SNR is [14]
(
Clo ≈ B log2
Pt
2
𝜎noise
)
𝜆2max
bps
(10.33)
where 𝜆max is the largest singular value.
Example
A 4 × 4 MIMO system has Pt = 1 W and an SNR of 4 dB. Find the power allocation using waterfilling.
⎡− 0.7471
⎢ 0.7695
H=⎢
⎢ 0.7849
⎢
⎣− 0.0523
0.9534
0.4432 0.3454⎤
0.3823 1.0413 0.3896⎥
⎥
0.5910 0.1215 0.5153⎥
⎥
− 0.0454 − 0.2182 1.2589⎦
Solution
2
= Pt ∕10SNR∕10 = 1∕104∕10 = 0.398
The noise variance is 𝜎noise
Use MATLAB to find the SVD (Table 10.1).
[U,D,V] = svd(H);
Table 10.1 Water filling example.
Subchannel
1
𝜆
2
∕𝜆2n
1∕SNRn = 𝜎noise
2
1.7889
1.3038
3
4
1.2000
0.5477
0.1244
0.2341
0.2764
)
(
4
∑
2
1∕SNR0 = Pt +
𝜎noise
∕𝜆2n ∕4 = 0.7404
1.3267
n=1
Is 1/SNR0 − 1/SNRn > 0
Yes
Yes
No
—
set P4 = 0
(
1∕SNR0 =
Yes
Pt +
3
∑
n=1
)
2
𝜎noise
∕𝜆2n ∕4 = 0.5450
Is 1/SNR0 − 1/SNRn > 0
Yes
Yes
Yes
Final power allocation
1
1
−
SNR0 SNRn
0.4206
0.3109
0.2686
The transmit power is allocated 42% to subchannel 1, 31% to subchannel 2, 27% to subchannel 3,
and 0% to subchannel 4.
359
360
10 MIMO
MIMO capacity equals the sum of all the subchannel capacities when the
transmit power is optimally spread across the N k subchannels [13].
C=
max
Pn ,
10.3.3
Nk
∑
n=1
Nk
∑
2
B log2 (1 + Pn 𝜆2n ∕𝜎noise
) bps
(10.34)
Pn ≤P n=1
CSIR and No CSIT
When there is no CSIT, then the optimal transmit weights cannot be found, so
setting all the transmit weights to one makes the most sense. The first step in
finding the receive weights is to express the signal in terms of power [15]
P = s†r sr = s†t HH† st = s†t CH st
(10.35)
Now, the N r × N r covariance matrix CH = HH† is square and an eigenvalue
decomposition is possible:
⎡𝜆1 0 0 ⎤
⎥
⎢
CH = Q ⎢ 0 ⋱ 0 ⎥ Q−1
⎢0 0 𝜆 ⎥
⎣
Nr ⎦
(10.36)
where the columns of Q are the eigenvectors and 𝜆n are the eigenvalues.
Eigenvalue n corresponds to the received signal power level in eigenchannel
(subchannel) n. The off-diagonal elements of CH are the correlation between
the transmitted signal streams. An increased correlation results in a decreased
capacity.
When the transmitter does not know channel characteristics, each element
receives an equal share:
Pn = Pt ∕Nt
(10.37)
Uncorrelated transmit elements in a random channel have an average capacity given by [8]
[ (
)]}
{
Pt
†
bps
(10.38)
HH
C = E B log2 det INt +
2
Nt 𝜎noise
MIMO capacity increases linearly with the number of elements for an equal
number of transmit and receive antennas. N t should be of the order 2N r [1].
When N t and N r are large and N t > N r , the capacity is [8]
C = Nr B log2 (1 + Nt Pt ∕Nr ) bps
(10.39)
As long as the ratio of N t /N r is constant, the capacity is a linear function of
Nr.
Problems
The instantaneous eigenvalues of a random CH have limits defined by [7]
√
√
√
√
( Nt − Nr )2 < 𝜆n < ( Nt + Nr )2
(10.40)
When N t is much larger than N r , all the eigenvalues cluster around N t . Each
eigenvalue is nonfading due to the high-order diversity when there are a large
number of transmit elements. Thus, the uncorrelated asymmetric channel with
many antennas has a very large theoretical capacity of N r equal, constant channels with gains of N t .
Example
For the following matricies, find (a) condition numbers (b) determine if (10.40)
is true.
⎡0.6948 0.0344 0.7655⎤
⎡1.0000 1.1000 1.2000⎤
H1 = ⎢0.3171 0.4387 0.7952⎥ H2 = ⎢2.0000 2.0000 2.0000⎥
⎢
⎢
⎥
⎥
⎣0.9502 0.3816 0.1869⎦
⎣3.3000 3.0000 3.1000⎦
Solution
(a) cond(H1 ) = 4.5408 and cond(H2 ) = 113.0466
(b) For this channel matrix, N t = 3 and N r = 3, so 0 < 𝜆n < 12
H1 is a random matrix and has eigenvalues given by
eig(H1′ ∗ H1).′ = 0.1238 0.4018 2.5531
All of these eigenvalues fall between 0 and 12
H2 is not a random matrix and has eigenvalues given by
eig(H2′ ∗ H2).′ = 0.0035 0.0402 45.1063
The high condition number of H2 indicates that there is a correlation between
rows or columns.
Problems
10.1
Find the SVD of the following channel matrices:
⎡0.2 0.4 0.8⎤
⎢
⎥
(a) ⎢0.7 0.3 0.4⎥ ,
⎢0.5 0.7 0.2⎥
⎣
⎦
⎡−0.6
⎢ 0.7
(d) H = ⎢
⎢ 0.7
⎢
⎣−0.1
[
(b)
0.1 0.5 0.7 0.9
0.2 0.6 0.4 0.3
0.7 0.3 0.4⎤
0.4 0.9 0.5⎥
⎥
0.6 0.1 0.5⎥
⎥
0.0 0.1 0.8⎦
]
,
(c)
]
[
0.4 0.5
0.8 0.2
,
361
362
10 MIMO
10.2
Find the eigenvalues of H† H for the channel matrices in Problem 10.1.
10.3
Find the condition number of the matrices in Problem 10.1.
10.4
A 4 × 4 MIMO system has Pt = 1 W W and an SNR of 5 dB. Find the
power allocation using waterfilling
⎡−0.6064
⎢ 0.7629
H=⎢
⎢ 0.7129
⎢
⎣−0.1070
0.7356 0.3441 0.4011⎤
0.4043 0.9241 0.5388⎥
⎥
0.6188 0.1806 0.5181⎥
⎥
0.0747 0.0611 0.8884⎦
10.5
Redo Problem 10.4 when the SNR is 20 dB.
10.6
Redo Problem 10.4 when the SNR is 2 dB.
10.7
A 3 × 3 MIMO system has CSIR but no CSIT. Find the capacity if
SNR = 10 dB, B = 1 kHz, and
⎡ 0.4508 0.5711 0.3450 ⎤
⎢
⎥
H = ⎢−0.2097 0.4704 0.4510 ⎥
⎢−0.6134 −0.6382 −0.4621⎥
⎣
⎦
10.8
Repeat Problem 10.7 using SVD.
10.9
What is the capacity if the lowest SNR subchannel in Problem 10.7 is
ignored?
10.10
Use waterfilling to allocate the power in Problem 10.7 when the total
power is 1 W, the noise power is 0.1 W and the signal bandwidth is
50 kHz. What is its new channel capacity?
References
1 Winters, J.H. (1987). On the capacity of radio communication systems with
diversity in a Rayleigh fading environment. IEEE Journal on Selected Areas
in Communications 5: 871–878.
2 Foschini, G.J. (1996). Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas. Bell
System Technical Journal 1 (2): 41–59, Autumn.
References
3 Agilent Technologies (2008). MIMO wireless LAN PHY layer [RF] opera-
tion & measurement, Application Note 1509, 29 April 2008.
4 Rohde&Schwarz, Introduction to MIMO, Application Note 1MA102.
5 Laurenson, D. and Grant, P. (2006). A review of radio channel sounding
6
7
8
9
10
11
12
13
14
15
techniques. In: 14th European Processing Conference, Florence, Italy, 1–5.
IEEE.
Proakis, J.G. and Manolakis, D.G. (2007). Digital Signal Processing Principles, Algorithms, and Applications, 4e. Upper Saddle River, NJ: Pearson
Prentice Hall.
Bliss, D.W., Forsythe, K.W., and Chan, A.M. (2005). MIMO wireless communication. Lincoln Laboratory Journal 15 (1): 97–126.
Agilent Technologie 2010. MIMO channel modeling and emulation test
challenges, Application Note, 22.
Jensen, M.A. and Wallace, J.W. (2004). A review of antennas and propagation for MIMO wireless communications. IEEE Transactions on Antennas
and Propagation 52 (11): 2810–2824.
Molisch, A.F. (2011). Wireless Communications, 2e. West Sussex, UK: Wiley.
Andersen, J.B. (2000). Array gain and capacity for known random channels
with multiple element arrays at both ends. IEEE Journal on Selected Areas
in Communications 18 (11): 2172–2178.
Telatar, I. E. (1996). Capacity of multi-antenna Gaussian channels. Tech.
note, AT&T Bell Lab.
Goldsmith, A. (2005). Wireless Communications. New York: Cambridge
University Press.
Brown, T., De Carvalho, E., and Kyritsi, P. (2012). Practical Guide to the
MIMO Radio Channel with MATLAB Examples. West Sussex, UK: Wiley.
Browne, D.W., Manteghi, M., Fitz, M.P., and Rahmat-Samii, Y. (2006).
Experiments with compact antenna arrays for MIMO radio communications. IEEE Transactions on Antennas and Propagation 54 (11): 3239–3250.
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Security
Individuals, companies, and governments demand data security from wireless
networks to protect their privacy. Untethered/wireless access to a network
gives users tremendous freedom of movement but requires more security
measures than a wired network. To obtain access to a wired network, an unauthorized user must physically connect to a port. In contrast, an unauthorized
user only needs to be within range of an antenna to connect to a wireless
port. Security means protecting communication and computing services,
information and data, personnel, and equipment for customers, government,
and network providers [1].
The Internet consists of a mix of wired and wireless systems with many
opportunities for security breaches (Figure 11.1). Wireless communication and
security requirements depend on the connectivity of their fixed and mobile subsystems [2]. This chapter introduces wireless security vulnerabilities and ways
to mitigate them. In order to understand the threats to wireless security and
how to defend against them, this chapter starts with background about wireless
networks and how devices connect to them. The second half of the chapter
deals with defenses against the treats with special emphasis on encryption.
11.1 Wireless Networks
A group of devices communicate with each other through a network. A local
area network (LAN) connects devices to a server over a common communications link. If the link is wireless, then the LAN is a wireless local area network
(WLAN). This section introduces the basics of a wireless network.
11.1.1
Addresses on a Network
Communication between two people at a distance requires location information like an address or a telephone number. Just like people, wireless network
devices find each other using their addresses on the network. The network
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
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11 Security
Internet/IPv6
WAN – 3GPP
2G,3G,4G – LTE, 5G
LPWAN
SigFox, LoRa
LAN, WiFi
ZigBee
PAN
Bluetooth
NFC
ZigBee
GPS
Figure 11.1 Various types of wireless networks, hierarchically connected. Source: Burg et al.
[3]. Reproduced with permission of IEEE.
interface card (NIC) connects a computer to the Internet. Every NIC has a
hardware address assigned by the manufacturer called an extended unique
identifier (EUI) or more commonly a MAC (media access control) address [4].
A NIC converts data into a signal, then transmits it in a packet over the network. All equipment connecting to computer networks (computers, routers,
servers, printers, smartphones, etc.) have a MAC (EUI) address. A 48-bit EUI
(EUI-48) address has 12 hexadecimal characters broken into two 24 bit codes
00 1f 19 ba 20 39
⏟⏞⏞⏟⏞⏞⏟ ⏟⏞⏞⏟⏞⏞⏟
OUI
(11.1)
NIC specific
This 48-bit address provides up to 248 = 281, 474, 976, 710, 656 unique values
for locating all devices connected to the Internet. EUI/MAC addresses serve
to direct packets from one device to another on a network. The proliferation
of devices connecting to the Internet forced the adoption of an updated 64-bit
EUI (EUI-64) which allows 264 unique addresses.
An EUI address is either a universally administered address (UAA) or a
locally administered address (LAA). Manufacturers assign unique UAAs to
devices. A network administrator has the ability to replace the UAA with
an LAA, so the EUI becomes the new LAA. The first 24 bits in (11.1), the
organizationally unique identifier (OUI), indicate the specific vendor for
that device. An “assignee” (vendor, manufacturer, or other organization)
purchases an OUI from the Institute of Electrical and Electronics Engineers
(IEEE) Incorporated Registration Authority. For instance, Apple has the OUI
“FCFC48” [5]. Table 11.1 has an example of the OUI having a base-16 value
of “ACDE48” which corresponds to the octet representation “AC-DE-48.” The
last row contains the binary representation.
Each device connected to a computer network has an IP (Internet protocol)
address assigned by the network provider. The IP address serves as a network
interface identification and a location address. Static IP addresses remain the
11.1 Wireless Networks
Table 11.1 Example OUI.
Octet identifier
0
Hexadecimal
Binary
1
AC
1010
2
DE
1100
1101
48
1110
0100
Figure 11.2 Example of an IPv4
address.
1000
172.32.254.2
10101100.00100000.11111110.00000010
same, while dynamic IP addresses change. A data packet has an origination IP
address and a destination IP address. The original Internet protocol version 4
(IPv4) address had 32 bits and is still in use (Figure 11.2). The current IPv6 has
128-bit IP addresses to accommodate the large number of devices connected
to the Internet.
The Internet uses EUI addresses in combination with IP addresses to route
packets from an origin to a destination – even within small LANs where two
computers communicate directly with each other. FedEx and UPS operate in
an analogous manner to the Internet. A FedEx or UPS worker does not carry
a package directly from the sender to the recipient. Instead, the package goes
to a sorting office. From there, it travels to several intermediate facilities that
redirect it to the final delivery at the destination. Routers behave like the various facilities that handle the package from the origin to the destination. An IP
packet temporarily stops at many different routers as it travels over the Internet. When a router receives a packet, it sends the packet to the next stop based
on the destination EUI address rather than the destination IP address. It strips
off the old destination EUI address (which was the router’s own EUI address)
and replaces it with a new destination EUI address that points to the next router
along the way to the final IP address. The packet passes from router to router
until it arrives at the final destination. Figure 11.3 has an example of a computer
sending data to another computer via the Internet. Note how the EUI addresses
in the packets change but the source and destination IP addresses do not.
11.1.2
Types of Wireless Local Area Networks
A wireless network has nodes (e.g. routers in Figure 11.3) that communicate
data from a transmitter to a receiver. Figure 11.4 has diagrams of the following
important network topologies for node communication:
(1) Star network: Nodes do not communicate with each other. All nodes
directly communicate with the base station. This network increases the
367
IP address: 195.15.16.11
EUI address: 00:1f:19:ba:20:39
IP address: 2.17.169.198
EUI address: 01:53:aa:f9:d2:6c
Data
Data
Data broken
into packets
Packets reassembled
into original data
Router
Transmitter
Source IP: 195.15.16.11
Destination IP: 2.17.169.198
Source EUI: 00:1f:19:ba:20:39
Destination EUI: 28:18:78:5a:f4:c7
Data
packet
Data
packet
Router
Data
packet
Data
packet
Data
packet
Receiver
Source IP: 195.15.16.11
Destination IP: 2.17.169.198
Source EUI: 35:a0:b1:00:57:c2
Destination EUI: 01:53:aa:f9:d2:6c
Router
IP address: 203.19.3.57
EUI address: 28:18:78:5a:f5:96
IP address: 2.17.169.1
EUI address: 35:a0:b1:00:57:c2
Source IP: 195.15.16.11
Destination IP: 2.17.169.198
Source EUI: 28:18:78:5a:f5:96
Destination EUI: 35:a0:b1:72:01:19
Figure 11.3 Data transfer between two computers on the Internet.
11.1 Wireless Networks
Star network
Mesh network
Tree network
Cellular network
Figure 11.4 Overview of main network topologies. Source: Burg et al. [3]. Reproduced with
permission of IEEE.
risk of a network failure, increases latency, and incurs a potentially large
overhead that degrades network capacity but is very simple.
(2) Tree networks: Nodes communicate with a designated neighboring node
that forwards the data traffic to a destination. This approach extends the
range of each node while routing occurs through adjacent nodes.
(3) Mesh networks: A flexible, robust network that allows nodes to connect to
other nodes with shorter latency and more system capacity but increases
routing complexity.
(4) Cellular networks: A cellular network is a star topology that has multiple
star networks arranged in a way that minimizes the coverage overlap of the
base stations. Routing data between base stations is handled via a separate
network. The infrastructure is complex and costly but has a high network
capacity.
A cyber-physical system (CPS) collects sensor and actuator data in order to
monitor the physical environment and analyze how the changes impact their
operation. The CPS then autonomously (sometimes with human-in-the-loop)
influences the physical environment [6]. The IoT (Internet of things) interconnects smart devices and connects to CPSs. Examples of CPSs include [7]:
large-scale environmental systems (e.g. natural resource management), power
and energy generation and distribution, transportation infrastructure, home
automation, autonomous driving, personal healthcare, logistics, or industrial
manufacturing. Highly distributed CPSs require wireless communications. An
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OT (operational technology) system is a CPS that supports the operation of an
industrial control processes. A wireless CPS has problems with latency, range,
throughput, power consumption of the node, and security.
11.1.3
WLAN Examples
An autonomous vehicle has a wireless vehicle-to-infrastructure communication system to obtain traffic updates and let car manufacturers monitor vehicle
status over the Internet (Figure 11.5). Vehicle-to-vehicle communication
systems use high speed, reliable links that provide traffic status or link to other
autonomous vehicles on the road. Cars have several independent CPSs that
overlap, such as anti-lock braking, adaptive cruise control, and automated
temperature control. These functions monitor actuators and sensors (e.g. tire
pressure, temperature, crankshaft position, light, and collision sensors). Most
CPS control and data communication occurs over wires in order to guarantee
integrity and 100% availability.
In 2014, two researchers hacked into a 2014 Jeep Cherokee through a wireless
entertainment and navigation system called Uconnect by knowing the vehicle’s
IP address [8]. They remotely gained access to a Jeep Cherokee’s controller area
network in order to take control of the vehicle, including shutting it down. This
hack prompted Fiat Chrysler to recall 1.4 million vehicles.
Smart
device
GPS
AM/FM
On-board
sensors
network
Remote key
(proprietary)
V2X
(802.11p)
In-vehicle network
Figure 11.5 Example of wired and wireless communications in an autonomous vehicle.
Source: Burg et al. [3]. Reproduced with permission of IEEE.
11.1 Wireless Networks
Sensor
gateway
Body sensor
network
WiFi, ZigBee
Bluetooth LE
3GPP, LTE
Diagnostics
Automated
medication
Control
Remote
server
Figure 11.6 Implantable personal medical devices communicate with wireless devices that
send data to the Internet. Source: Burg et al. [3]. Reproduced with permission of IEEE.
Figure 11.6 shows an example of wireless CPS in a personal healthcare system.
Implanted medical devices perform and monitor biological functions such as
heart rate and glucose level. Sensors record vital data then wirelessly transmit it
to a receiver (e.g. smartphone) that in turn connects to the Internet. Health professionals anywhere in the world can monitor patient status in order to diagnose
and advise patients. The system requires stringent security in order to prevent
life threatening intrusions and protect patient privacy.
Figure 11.7 presents an example of a blood glucose monitoring system.
A sensor mounted on the abdomen measures the blood glucose. Every five
minutes, a transmitter attached to the sensor sends the data to a dedicated
receiver or smart phone via Bluetooth (Appendix D). These devices store
the data for 24 hours then uploads the data to a website via the Internet. The
website maintains a database accessible by authorized people. Patients want
this type of data secured.
Figure 11.8 portrays a smart home system that has small-scale CPSs, such
as HVAC (heating, ventilation, and air conditioning). The CPSs connect to
a central hub through a short-range, latency insensitive, and fault-tolerant
wireless systems. Public services or energy suppliers sometimes have access to
autonomous sensors distributed throughout the house. Xcel Energy offers the
voluntary Savers Switch program that compensates a homeowner for letting
Xcel install a switch on the air conditioners that it controls. On hot summer
days, the company cycles the air conditioning every 15–20 minutes in order to
conserve power usage at peak times of the day.
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Bluetooth to
smartphone
Transmitter and sensor
attached to person
Internet
connection to
company
website
Figure 11.7 Wireless glucose monitoring system.
Z-Wave
LoRa
Thermostat
Smart meter
ZigBee
Sensor
Z-Wave
Light control
Bluetooth
Access control
IR/VLC
Infotainment
Z-Wave
HVAC Control
WiFi
Surveillance
Wi-Fi, LTE, 3GPP, Broadband
Figure 11.8 Smart home connectivity. Source: Burg et al. [3]. Reproduced with permission of
IEEE.
11.2 Threats
Since wireless communication systems, like these examples, pass important
information and commands, they need protection from unauthorized users.
The large number of wireless devices in the home increase radio frequency
interference (RFI) and potential security breaches.
11.2 Threats
Wireless devices join a WLAN through an access point (AP). The AP connects to a wired LAN typically through Ethernet. Ethernet sets the packet format that other devices on the LAN recognize, receive, and process. Figure 11.9
shows a wireless router connected to a cable provider box for Internet access
through a coaxial cable. The router antennas receive the signal from a wireless
device. Next, the AP inside the router transfers the data to the Internet. Routers
have multiple input Ethernet ports in order to connect to devices over Ethernet
cables.
A service set consists of wireless network devices that communicate with
each other through the same network [9]. The three types of service sets are
Basic Service Set (BSS), Independent Basic Service Set (IBSS) or ad hoc network, and Extended Service Set (ESS) [10].
Antenna
Input
ethernet
cable
Router
Power
Power
Output
ethernet
cable
Figure 11.9 Wireless router.
Coax
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A device gains access to the network through the AP by meeting the following
criteria [11]:
• A matching Service Set Identifier (SSID)
• A compatible wireless data rate
• Authentication credentials.
Once the device connects to the AP, all communications associated with the
device pass through the AP. An AP covers the Basic Service Area (BSA) much
like a base station covers a wireless cell. The AP usually has a wired connection
to a larger network. Every AP has a 32-bit SSID or network name that uniquely
distinguishes it from other APs in the same physical area. The SSID acts like a
password that devices need to access the WLAN.
IBSS allows multiple devices to communicate directly with each other rather
than through a central device. This peer-to-peer network between devices does
not require an AP, so it cannot connect to a BSS. An IBSS is ad hoc, because it
does not need APs and routers.
An ESS expands the WLAN coverage area by connecting multiple BSSs
through a distribution system (wired or wireless) as shown in Figure 11.10. A
device that moves (roams) from one BSS to another remains connected to the
LAN. When the signal gets weak, a device switches from one AP to another
AP that has a better link. An ESS usually has an SSID that allows roaming from
one AP to another without requiring the device to reconfigure. The ESS covers
an extended service area (ESA) that consists of all the BSAs within its control.
A WLAN threat occurs when an attacker obtains unauthorized access. An
attacker steals data or gains control of a system by entering an attack surface.
The attack surface consists of all points where an attacker enters a system. An
attack surface of an application consists of [12]
• The sum of all data/command paths that communicate with an application.
Distribution
system
AP
BSS
AP
AP
ESS
Figure 11.10 An ESS consists of multiple BSSs connected through a distribution system.
11.2 Threats
• The code that protects data/command paths, such as resource connection
and authentication, authorization, activity logging, data validation, and
encoding.
• All valuable application data, including secrets and keys, intellectual property, critical business data, and personal data.
• Data protection, such as encryption and checksums, access auditing, and
data integrity and operational security controls.
Many different kinds of threats prey on WLANs via an attack surface.
Microsoft developed a model called STRIDE that places threats into six
categories [13]:
• Spoofing is when one user poses as another user or administrator in order
to fraudulently obtain authentication information, such as a username and
password.
• Tampering means that the attacker modifies data. An example is changing
an account balance.
• Repudiation means to delete or alter a login or transaction data. An example
is deleting a purchase transaction to avoid payment.
• Information disclosure refers to stealing sensitive information (e.g. proprietary or secret data).
• Denial of service (DoS) attacks overwhelm a system, so that nobody can
access the network.
• Elevation of privilege occurs when an unprivileged user obtains privileged
access to the entire system. The attacker bypasses system defenses and
becomes trusted.
Threats attack wireless networks in seven ways [14]:
1. An insertion attack means that an unauthorized wireless device joins a BSS.
2. A misconfiguration attack occurs when software is not properly setup or
updated. Devices gain access through default SSIDs or through brute force
guessing an SSID.
3. Wireless sniffing tools capture the initial part of a wireless connection that
includes a username and password in an interception attack. A device then
enters the AP as a valid user.
4. War driving attacks result from a mobile device that searches for and then
exploits WLANs.
5. A RAP (rogue AP) connects to a network without authorization from an
administrator. RAPs have proliferated due to low-cost hardware that appears
invisible to the legitimate WLAN.
6. In a client-to-client attack, one user attacks another user on the same
BSS/ESS.
7. A jamming attack denies user access to a WLAN by overwhelming the AP
with interfering signals.
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11.3 Securing Data
A code maps one group of symbols into a new group of symbols using a
codebook that contains all the mappings. For instance, the ASCII codebook
maps a letter into bits. In contrast, a cipher transforms one symbol into a
new symbol using an algorithm. For instance, the symbol 110 converts to 111
using the algorithm add 001 to the symbol. Codes are relatively easy to break,
and the codebooks are difficult to distribute and keep secure. Consequently,
ciphers have become the primary way to secure data.
11.3.1
Cryptography
Cryptography transforms (encrypts) plaintext (message or data) into ciphertext
(an unreadable format that masks the content) using a cipher (algorithm). It
hides the meaning of a message but does not hide the existence of the message
[15]. Cryptography has five primary functions [16]:
• Privacy/confidentiality: Ensuring that no one reads the message except the
intended receiver.
• Authentication: Verifying user identity.
• Integrity: Assuring the receiver that the received message is the same as the
transmitted message.
• Nonrepudiation: Proof of the sender’s identity.
• Key exchange: The sharing of crypto keys between sender and receiver.
Cryptography started over 2000 years ago when Julius Caesar invented the
Caesar cipher that shifts letters in the alphabet a set number of places known
to the sender and receiver (in Caesar’s case three) [17]. The 25 distinct shift
ciphers for a 26-letter alphabet made this code relatively easy to break. Arab
scholars eventually cracked codes that simply substituted one letter for another,
by noting that some letters occur more often than others in written documents.
Frequency analysis of letters in the English language indicate that “e” occurs
most often (see Figure 1.6). Thus, any alphabetic substitution cipher in English
has the most common letter in the code representing “e.”
In 1882, Frank Miller developed an unbreakable encryption method that
was later called one-time pad encryption [18]. One-time pad encryption
converts data into meaningless characters using a pseudo random noise (PRN)
generator to determine symbol substitution. Both sender and receiver need the
same substitution algorithm and PRN generator. Using PRNs to create rules
for substituting one symbol for another makes the one-time pad encryption
difficult to break using frequency analysis. Figure 11.11 is an example of a
one-time pad encryption using PRNs generated by Google to encode the
message “HELLO THERE.”
11.3 Securing Data
Figure 11.11 Example of one-time pad
encryption. Source: Abellán and Pruneri
[18]. Reproduced with permission of
IEEE.
H
E
L
L
O
T
H
E
R
E
21 14 21
9
16
23 18
6
1
10
C
U
E
Q
K
S
O
S
G
Z
Figure 11.12 Enigma machine at the National Cryptologic Museum.
The famous Enigma cipher was initially developed by the Dutch for banking
communications. Germans bought the patent in 1923 and created an electromechanical machine that substituted one letter for another as a message
was typed (Figure 11.12). Each day, the electrical and mechanical connections
for each machine were changed according to rules that were distributed
once a month. Enigma had approximately 159 quintillion different settings
[19]. Enigma would have been unbreakable if the users had followed proper
operating procedures and spies had not passed on critical information. In
1932, Polish cryptanalysts decoded German Enigma ciphers [20]. They were
successful in breaking the ciphers and producing their own Enigma machines.
In 1939, Poland shared their breakthroughs with France and the United
Kingdom. Alan Turing with a team of scientists discovered that a letter can be
encrypted as any letter other than itself. In addition, the Germans put “Heil
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11 Security
Hitler” at the end of every message. That hint provided enough information to
crack the code (Figure 11.12).
Cryptography creates encrypted data known as ciphertext (Ctext ) from
unencrypted data known as plaintext (Ptext ) by applying mathematical transformations that converts the data into a secret code [21]. Only authorized
users know the algorithm that unencrypts the ciphertext. Deciphering applies
an inverse mathematical transformation to the secret code in order to recover
the original data.
Encription ∶ Ctext = (Ptext )
(11.2)
Decription ∶ Ptext =  −1 (Ctext )
(11.3)
where  = encryption and  = decryption. In theory, unauthorized users
cannot access the original data without knowing  −1 .
A key is a random string of bits that scrambles and unscrambles data. Keys
should be long, unpredictable, and unique. Three categories of data encryption
based on the use of a secret key are [16]:
−1
• Secret key cryptography (SKC) or symmetric encryption uses the same key
for encryption and decryption.
• Public key cryptography (PKC) or asymmetric encryption uses one key for
encryption and a second key for decryption.
• Hashing uses a mathematical transformation to irreversibly encrypt data and
provide a digital fingerprint.
Figure 11.13 differentiates between the types of encryption. Note that the
hash function is one-way encrypts plaintext into ciphertext that is nearly
impossible to decrypt.
Plaintext
Once upon
a time...------------------------------------------------------------
Ciphertext
Encryption
zaq12wsxy
de345rtfgv Decryption
bhny678uij
km,.lo0=][
p;’/.,mnfde
3q=wpj6ml
Plaintext
Once upon
a time...------------------------------------------------------------
(a) SKC: symmetric
(b) PKC: asymmetric
(c) Hashing
Figure 11.13 Basic functioning of the three categories of encryption.
11.3 Securing Data
11.3.2
Secret Key Cryptography
SKC encrypts the plaintext with a key then sends the ciphertext to the receiver.
The receiver uses the same key to decrypt the ciphertext and recover the
plaintext. Security depends on the difficulty of guessing the key. Examples of
SKC algorithms include the triple data encryption standard (3DES) and the
advanced encryption standard (AES). SKC creates either stream ciphers or
block ciphers.
Stream ciphers operate on one bit of plaintext at a time with a key of
pseudorandom bits to create ciphertext much longer than the plaintext. Using
an unpredictable PRN generator and one-time keys improves SKC security.
Stream ciphers approximate the one-time pad cipher.
Block ciphers encrypt N bits of data (block) at one time. Blocks usually contain 64, 128, or 256 bits [22]. The most important operating modes of a block
cipher are [23]:
• Electronic Codebook (ECB) encrypts a plaintext block into ciphertext block
using a secret key. The same plaintext block always encrypts to the same
ciphertext block. ECB is susceptible to many forms of attack. A single bit
error in the ciphertext causes errors in the entire block of decrypted plaintext.
• Cipher Block Chaining (CBC) performs an exclusive or of the plaintext with
the previous ciphertext block before doing the encryption. In this way, two
identical plaintext blocks have different encryptions. CBC protects against
most brute-force, deletion, and insertion attacks. One bit error in the ciphertext, however, causes errors in the entire decrypted plaintext block as well as
a bit error in the next decrypted plaintext block.
• Cipher Feedback (CFB) encrypts data into groups of bits smaller than the
block size. A single bit error in the ciphertext affects the current and next
block.
• Output Feedback (OFB) generates the keystream independently of both the
plaintext and ciphertext bitstreams. A single bit error in OFB ciphertext generates one bit error in the decrypted plaintext.
• Counter (CTR) mode uses different keys for different blocks so that two identical blocks of plaintext will not result in the same ciphertext. Each block of
ciphertext has a specific location within the encrypted message. CTR mode
processes blocks in parallel – thus offering performance advantages when
parallel processing and multiple processors are available – and is not susceptible to ECB’s brute-force, deletion, and insertion attacks.
11.3.3
Public Key Cryptography
PKC encrypts the data with one key and decrypts it with a different key [24].
One of the keys is private and only known to the user, while the other key is
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public and known to others. The sender encrypts the information using the
receiver’s public key. The receiver decrypts the message using a private key.
The receiver knows who sent the message (authentication), and the transmitter
cannot deny sending the message (nonrepudiation). PKC has easy-to-compute
one-way functions for creating the cipher, but the decryption inverse function
is difficult to compute. In fact, SKC decrypts a message about 1000 times faster
than PKC, so PKC is not used for message encryption [24]. The cipher and
decipher keys are mathematically related, but knowing one key does not lead
to the other key. PKC has the significant advantage over SKC of having no key
distribution.
11.3.4
Hashing
Hashing transforms a character string via a hash function into a finite value
called a hash [25]. The hash quickly locates an item in a database via a hash
table. Two simple examples of hashes are
• Student identification numbers used to retrieve private information about a
student.
• Book call numbers used to quickly locate books in a library (e.g. Library of
Congress Classification).
Unlike SKC and PKC, hashing performs one-way encryption. Hashing has
three steps:
1. Convert data into a hash using a hash function.
2. Store the data in a hash table.
3. Quickly retrieve the data from the table using the hash.
A hash function generates a number (hash) for an object or data string. Two
equivalent objects have the same hash while two unequal objects do not. A
collision occurs when two different objects have the same hash. A hash function
should [26]
1. Be easy to compute.
2. Uniformly distribute storage across the hash table.
3. Avoid collisions.
Example
Map the data D = [54 26 93 17 77 31] into a hash table with N = 11 slots. The
hash function performs modulo 2 arithmetic mod(D,N) to get the hashes. Find
the hashes for the elements in D and create the hash table.
Solution
Use the MATLAB command: hash = mod([54 26 93 17 77 31],11)
11.4 Defenses
hash = 10
4 5 6 0 9
Hash
0
1
2
3
4
5
6
7
8
9
10
Data
77
—
—
—
26
93
17
—
—
31
54
Hash functions ensure that a file remains unchanged – ideal for guaranteeing
data integrity. Any change made to a message causes the receiver to calculate a
hash value different from the one transmitted. A good hash function does not
produce the same hash value for different inputs. Small changes to the input
string of the highly nonlinear hash function produce a big change in the hash.
Hashing verifies a string’s identity by comparing it with a securely stored string.
For instance, using the last four digits of a credit card assists in verifying someone’s identity. A user gains access to a network by comparing the login password
to the stored hash. Hashing excels at storing passwords, because even administrators cannot decrypt the hash to gain access to the passwords. Hashing is
more secure than encryption and should always be used unless the cybertext
needs decrypted [25].
11.4 Defenses
Section 11.3 established the vulnerabilities associated with a wireless system.
This section covers some approaches to defend wireless systems against attacks.
Users expect that a message sent over a communication channel has [26]
• Authentication (the sender and receiver are accurately identified).
• Confidentiality (the message can only be understood by the receiver).
• Integrity (the message arrives unaltered).
A resilient system operates in spite of encountering unexpected inputs, subsystem failures, or environmental conditions that lie outside its operating range
[27]. Fault tolerance, fault detection, and adaptation enhance resilience.
Three approaches to securing a wireless network include [28]:
1. Requiring user authentication
2. Eliminating RAPs
3. Encrypting data.
Combining all three approaches provides the best security. All security measures require continuous updating otherwise, the system becomes vulnerable
to attacks.
Often times, the AP broadcasts its SSID, so it loses the basic security protection of SSID. A list of broadcasted SSIDs within range appears on a wireless
381
382
11 Security
device that attempts to connect to a Wi-Fi network. APs configured without an
SSID allow open access to any wireless device. A first step in securing a network
is to change default SSIDs and disable APs from broadcasting their SSIDs.
A second step in securing a WLAN filters EUI/MAC addresses and only
allows network access to an approved EUI address. EUI filtering works best
with small networks that frequently update the EUI addresses on the approved
list. EUI filtering blocks a hacker who hijacked a network IP address.
SSID and EUI address filtering satisfy the first two requirements of WLAN
Security. Encryption, the third defense, uses one of three protocols [29]:
• Wired Equivalent Privacy (WEP) is the original wireless encryption protocol designed to provide security equivalent to wired networks. WEP has
many security flaws, is difficult to configure, and is easily broken.
• Wi-Fi Protected Access (WPA) was a replacement for WEP while the IEEE
802.11i wireless security standard was being developed (Appendix E). Most
current WPA implementations use a preshared key, commonly referred to as
WPA Personal, and the Temporal Key Integrity Protocol (TKIP) for encryption. WPA Enterprise uses an authentication server to generate keys or certificates.
• Wi-Fi Protected Access version 2 (WPA2) is based on the 802.11i wireless
security standard that was finalized in 2004. WPA2 encrypts data with AES.
The U.S. government uses AES to encrypt top secret information.
The IEEE 802.11 standard specifies the RC4 SKC encryption algorithm with
40-bit or 104-bit keys for WEP. Concatenating a 24-bit “initialization vector”
with an encryption key produces new 64- or 128-bit keys. This key seeds a PRN
generator that creates a random sequence for encrypting the data. All clients
and APs using WEP on a wireless network have the same key for encrypting
and decrypting data. A client authenticates through a four-step process [21]:
1. Client requests authentication to the AP.
2. AP asks the client a challenge phrase.
3. Client encrypts the challenge phrase with the shared symmetric key before
transmitting it to the AP.
4. Client receives authorization when the client’s response matches the AP
challenge phrase.
In spite of its weaknesses, WEP provides protection for many home networks
and in small networks with low security requirements [30]. EUI address filtering
along with 128-bit WEP and SSID along with use of one-time keys establishes
reasonable security for many applications. WEP encryption/decryption slows
down data transmission.
A virtual private network (VPN) uses encryption to enable users to safely
access a secure private network over a public network like the Internet. The
11.4 Defenses
wireless client and the wireless network must have the VPN software installed.
A VPN is necessary for security on a public network accessible by anybody.
An IDS (intrusion detection system) monitors a network and identifies suspicious patterns indicative of an attack [31]. Like a burglar alarm on a house, an
IDS detects an intrusion but cannot prevent or respond to an intrusion. Two
main classes of IDS are rule based IDS and anomaly based IDS [2]. Rule-based
IDS (also called signature based IDS) detects intrusions by comparing data
with a list of signatures or patterns symptomatic of a malicious intruder. IDS
has few false positives and accurately detects well-known attacks. On the
down side, IDS misses attacks that do not have signatures in the intrusion
database and becomes slow in high data traffic. Anomaly based IDS looks for
abnormalities in data traffic patterns and “learns” patterns that correspond
to threats. Anomaly detection finds new threats without using a data base,
requires little maintenance, and becomes more accurate with time. On the
other hand, anomaly detection has a high false alarm when learning new
intruders during which it has false alarms.
Security measures are important during system design time as well as during
operation (runtime) [32]. Figure 11.14 delineates design time and runtime
security approaches for new and legacy systems. Design time methods verify
Threats
Design-time
attacks
Countermeasures
Threat analysis and hardening
Architectural
flaws
Component
flaws
Supplier
flaws
Safety-critical networked
control
QoS-aware serviceoriented architectures
Synthesis of control
software
Design time
Run time
Monitors
Run-time
attacks
Fingerprinting
Bugs
Repair
Architecture
Network
Modules
Figure 11.14 Safety and security measures at design time and runtime. Source: Wolf and
Serpanos [32]. Reproduced with permission of IEEE.
383
384
11 Security
subsystems and sets of subsystems properties. System designers scrutinize
system models to ascertain attack surfaces and safety issues before testing the
solution effectiveness. Runtime approaches (e.g. watchdogs, fingerprinting,
repair, etc.) protect against attacks and failures during system operation.
System characteristics that recognize early stage bugs and attacks can be
monitored at runtime.
Problems
11.1
Find an OUI for (a) Cisco Systems, Inc., (b) Apple, Inc., (c) Dell, Inc.,
and (d) Agilent Technologies, Inc.
11.2
Find the binary representation of these hexadecimal numbers:
(a) FC253F, (b) C8A70A, (c) 20C047, and (d) 948FEE.
11.3
Find hexadecimal representation of these binary numbers:
(a) 101001000100110011001000, (b) 11110000100010111000100,
(c) 11110000100010111000100, and (d) 110101001000000111010111.
11.4
Find the IPv4 and IPv6 addresses associated with a MAC address on a
computer. These are listed in the network properties on a computer.
11.5
Convert these decimal numbers to binary: (a) 9999, (b) 365, (c) 11111,
and (d) 9876.
11.6
Convert these binary numbers to decimal: (a) 111000111000111,
(b) 101010101010, (c) 11010011010000010111, and (d) 1111111111.
11.7
Find the company that has the following OUI: (a) 98 : 90 : 96,
(b) 08 : 00 : 20, (c) EC : AD : B8, and (d) B0 : AA : 77.
11.8
Use a Caesar cipher to encrypt the word “xray”.
11.9
Write MATLAB code that will decrypt a Caesar cipher with an arbitrary shift. Use this code to decrypt (a) “BNWJQJXX” with a five letter
shift and (b) “ZLUHOHVV” with a three letter shift.
11.10
Write a MATLAB function that does one-time pad encryption of the
capital letters of the English alphabet. Start the function with the following code:
function ncode=onetime(mess,key)
References
nm=length(mess);
nk=length(key);
alph='ABCDEFGHIJKLMNOPQRSTUVWXYZ';
Find the encryption from you m-file for:
(a) mess = ‘WIRELESS’; key = ‘XCVBNMASDF’ and
(b) mess = ‘WIRELESS’ key = ‘XCVB’
11.11
Write a MATLAB function to count the number of times that a letter
occurs in text. Distinguish between capital and lower case letters. Use
a stem plot to show your results.
function ltrfreq=letterfreq(text)
11.12
Find the hashes for the data below if the hash table has 23 slots:
[113 117 97
100 114 108 116 105 99]
References
1 (2015). Security in telecommunications and information technology.
2
3
4
5
6
7
8
In: ITU-T, Telecommunication Standardization Bureau (TSB) Place des
Nations – CH-1211 Geneva 20 – Switzerland, Sep 2015.
Liu, Y. and Zhou, G. (2012). Key technologies and applications of Internet
of Things. In: Proceedings of the 5th International Conference on Intelligent
Computing Technology Automation, 197–200.
Burg, A., Chattopadhyay, A., and Lam, K.Y. (2018). Wireless communication
and security issues for Cyber–Physical Systems and the Internet-of-Things.
Proceedings of the IEEE 106 (1): 38–60.
(2017). Guidelines for Use of Extended Unique Identifier (EUI), Organizationally Unique Identifier (OUI), and Company ID (CID). IEEE Standards
Association.
https://www.adminsub.net/mac-address-finder/apple (accessed 14 August
2018).
Shi, J., Wan, J., Yan, H., and Suo, H. (2011). A survey of cyber-physical
systems. In: Proceedings of the International Conference on Wireless Communications and Signal Processing, 1–6. IEEE Standards Association.
Khaitan, S.K. and McCalley, J.D. (2015). Design techniques and applications
of cyberphysical systems: a survey. IEEE Systems Journal 9 (2): 350–365.
Miller, C. and Valasek, C. (2015). Remote exploitation of an unaltered
passenger vehicle. http://illmatics.com/Remote%20Car%20Hacking.pdf
(accessed 30 July 2019).
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9 https://en.wikipedia.org/wiki/Service_set_(802.11_network) (accessed 1
October 2018).
10 https://www.certificationkits.com/cisco-certification/ccna-articles/cisco-ccna-
wireless/cisco-ccna-wirelss-bss-a-ess (1 October 2018).
11 Chapter 1 802.11 Network Security Fundamentals (2008). Cisco Secure
Services Client Administrator Guide, Release 5.1. Cisco Systems, Inc.
12 https://www.owasp.org/index.php/Attack_Surface_Analysis_Cheat_Sheet
(1 October 2018).
13 https://docs.microsoft.com/en-us/previous-versions/commerce-server/
ee823878(v=cs.20 (accessed 30 July 2019).
14 https://www.spamlaws.com/jamming-attacks.html (2 October 2018).
15 https://www.techopedia.com/definition/1770/cryptography (accessed 2
January 2019).
16 https://www.garykessler.net/library/crypto.html#intro (accessed 2 January
2019).
17 Singh, S. (1999). The Code Book: The Science of Secrecy from Ancient Egypt
to Quantum Cryptography. New York: Anchor Books.
18 Abellán, C. and Pruneri, V. (2018). The future of cybersecurity is the quan-
tum random number generator. IEEE Spectrum: 30–35.
19 https://www.scienceabc.com/innovation/the-imitation-game-how-did-the-
enigma-machine-work.html (4 October 2018).
20 https://www.cia.gov/news-information/blog/2016/who-first-cracked-the-
enigma-cipher.html (9 November 2018).
21 Geier, J. (2005). Wireless Networks First-Step. Indianapolis, IN: Cisco Press.
22 Feistel, H. (1973). Cryptography and Computer Privacy. Scientific American
228 (5): 15–23.
23 https://www.tutorialspoint.com/cryptography/block_cipher_modes_of_
operation.htm (accessed 28 January 2019).
24 Diffie, W. and Hellman, M. (Nov 1976). New directions in cryptography.
IEEE Transactions on Information Theory 22 (6): 644–654.
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between-hashing-and-encrypting (accessed 2 January 2019).
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availability-CIA (accessed 28 January 2019).
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measures and assessment. In: Resilience Assessment and Evaluation of
Computing Systems (eds. K. Wolter, A. Avritzer, M. Vieira and A. van
Moorseled). New York: Springer.
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January 2019).
29 Vacca, J.R. (2006). Guide to Wireless Network Security. New York: Springer.
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the Association for Information Systems 9: 269–282.
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January 2019).
32 Wolf, M. and Serpanos, D. (Jan. 2018). Safety and security in Cyber-Physical
Systems and Internet-of-Things Systems. Proceedings of the IEEE 106 (1):
9–20.
387
389
12
Biological Effects of RF Fields
Wireless systems expose people to a wide range of radio frequency (RF)
signals. The frequency and energy of these signals determine the extent
of their interaction with biological tissues. Gamma rays and X-rays have
well-documented deleterious effects on humans [1]. Some interesting but
nonharmful effects of electromagnetic fields occur at much lower frequencies. For instance, extremely low-frequency magnetic fields generated by
high-voltage power lines disrupt the geomagnetic field orientation of cattle
and roe deer while they graze [2]. Wireless communication systems typically
operate at frequencies between these two extremes. Constant exposure to
wireless communication signals motivates researchers to find any detrimental
impact on humans. This chapter introduces the interactions of RF radiation
from wireless communication systems with human biological functions.
12.1 RF Heating
Figure 12.1 divides the electromagnetic spectrum into ionizing and nonionizing radiation [3]. Ionization strips electrons from atoms and leads to tissue
damage. X-rays and gamma rays are examples of ionizing radiation with high
energy [4]. Gamma rays are at the pinnacle of the electromagnetic spectrum at a
frequency ≥3 × 1019 Hz (𝜆 ≤ 10−11 m) while X-rays lie one rung below with a frequency range of 3 × 1016 ≤ f ≤ 3 × 1019 Hz (10−11 ≤ 𝜆 ≤ 10−14 m). Wavelengths
close to the size of an atom (1 × 10−10 ≤ atom size ≤ 5 × 10−10 m) cause resonances that ionize atoms. Nonionizing RF radiation, on the other hand, does
not have enough energy to remove electrons from atoms [5].
Many research studies on RF interactions with tissue are controversial,
because they result from the analysis of data collected from a human population. These studies are difficult to control and rarely result in strong
correlations. Given the number of factors in any human population, even a
strong correlation does not prove a causation [6].
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
390
12 Biological Effects of RF Fields
Non-ionizing radiation
Low induced
currents
No proven effects
High induced
currents
Heating
RF
0
3
10
Ionizing radiation
106
Electronic
excitation
photochemical
effects
IR
109
Broken bonds
DNA damage
Light UV
1012
f (Hz)
10
15
X rays
1018
Gamma rays
1021
1024
Figure 12.1 Frequency spectrum and biological effects.
RF radiation heats tissue just like microwave ovens cook food. Microwave
ovens operate at 2.45 GHz which borders the cell phone frequency bands.
Many RF communication systems transmitting in the microwave band, such as
cell phones, heat nearby tissue. Cell phones cause health concerns like cancer,
because they transmit next to the human ear and head. Human tissue does
not dissipate excessive heat generated by high RF fields very well, especially in
areas of low blood flow, such as the eyes [7].
Low RF radiation levels produce insignificant body heating and have no
known harmful biological effects [6]. The search for nonthermal effects
continues without conclusive results. In some situations (e.g. working near
high-powered RF sources) appropriate limits ensure the safety of people in
the vicinity. Tissue, a lossy dielectric, heats via two primary mechanisms: ionic
conduction and dipolar polarization [8].
The heating conduction mechanism causes mobile charge carriers (electrons
and ions) to move back and forth through the material under the influence of
the microwave electric field, creating an electric current. These induced currents produce heating in the sample due to any electrical resistance resulting
from charges colliding with neighboring molecules or atoms.
Human tissue primarily consists of water. Tissue with the highest water content includes muscle, skin, liver, spleen, kidney, and brain, while low water content tissues include fat, bone, teeth, nails, and hair [9]. Water molecules have a
neutral charge; however, they are dipoles, because their electrons spend more
time around the larger oxygen nucleus than the small hydrogen nuclei, giving the oxygen end of the molecule a negative charge. Turning off the electric field causes the water molecules to return to random orientations in the
tissue as shown in Figure 12.2. Water dipoles want to align with the electric
field lines in their vicinity. Their ability to adjust orientation in order to align
with an applied electric field and release of thermal energy depends on frequency [10]. Low-frequency electric fields slowly change direction. The water
molecules respond to the field change with no delay as shown in Figure 12.3.
In contrast, Figure 12.4 shows the lack of response of the water molecules to
12.1 RF Heating
Figure 12.2 Random orientation of water molecules
with no electric field present.
−
+
O
H+
H
H+
O−
+
H+
H
H
H+
+
−
−
O
O
H
+
H+
O−
H+
O−
H+
H+
Electric field
H+
H+
H+
O−
H+
H+
H+
H+
O−
O−
O−
H+
H
H+
O−
H+
O−
H+
H+
O−
O−
H+
H+
+
O−
H+
H+
O−
H+
O−
H+
H+
O−
H+
+
H+
H
H+
Electric field
Figure 12.3 If the electric field varies too slowly, then the water molecules flip when the
field changes sign.
−
+
+
H
H
+
H+
H
H+
−
+
O
+
H
+
+
O−
H
+
H
H+
O−
−
O
−
O
H+
H
H
O− H+
H
O−
+
H
H+
O
+
H+
O−
Electric field
H+
H+
H+
H+
O−
H+
−
O
H+
H+
O−
Electric field
Figure 12.4 If the electric field varies too fast, then the water molecules do not have time to
change orientation.
391
12 Biological Effects of RF Fields
+
H
−
O
+
H
H+
+
H
+
H
−
O
+
H
+
H
+
H
−
O
+
+
H
H
+
−
H
O
+
H
−
O
+
H
−
O
H+
+
H
+
H
+
H
−
O
+
H
−
O
+
H
−
O
+
H
+
H
−
O
+
H
−
O
−
+
+
O
−
H
H
O
+
+
H
H
+
+
H
H
+
H
+
H
+
H
−
O
+
H
Electric field
−
O
+
H
O− H+
−
O
+
H
O−
+
H
−
O
+
H
+
H
−
O
H+
+
H
+
H
Electric field
O− H+
−
O
+
H
H+
392
H+
O−
+
H
H+
+
H
−
O
Figure 12.5 Water heating occurs when the field changes at the right speed that allows a
slow reorientation of the water molecules.
a very high-frequency electric field. The field changes much faster than the
water molecules have time to flip. A sweet spot in the spectrum flips the water
molecules with a delay. In this case (Figure 12.5), the random molecules align
with the applied electric field. Switching the field direction creates some delay in
the water molecule response. In the middle of the delay, the molecules become
randomly oriented again before aligning with the new direction of the electric
field. This constant change between a randomized state and an aligned state
produces heat. A microwave oven uses this concept to heat water molecules
in food.
Lossy dielectrics have a complex dielectric constant given by
𝜎
(12.1)
𝜀 = 𝜀′ − j𝜀′′ = 𝜀′ − j
2𝜋f
where 𝜎 is the conductivity. The dielectric heating power (W) due to an electric
field incident on a lossy dielectric is given by [11]
P = 2𝜋f 𝜀′′ tan 𝛿LF |E|2 Υ W
(12.2)
where tan 𝛿 LF = 𝜀′′ /𝜀′ is the dielectric loss factor, and Υ is the volume of the
dielectric. Even when an object has high dielectric loss, the heating efficiency
for a big sample is sometimes low due to the shallow penetration depth of
the microwaves. Consequently, the penetration depth quantifies the RF heating efficiency and distribution. The penetration depth (dp ) defines the distance
12.2 RF Dosimetry
from the surface to the point where the field strength drops by 1/e = 0.3679.
Neglecting magnetic effects, the penetration depth (same units as 𝜆) is given
by [12]
𝜆
dp = √
√
1
2𝜋[( 1 + tan2 𝛿LF − 1)] ∕2
(12.3)
Example
For muscle at 2 GHz: 𝜀′r = 53.3, 𝜎 = 1.45 S/m. Find the dielectric heating power
and dp when the electric field amplitude is 5 V/m.
Solution
𝜎
1.45
= 1.154 × 10−10
=
2𝜋f
2𝜋 × 2 × 109
1.154 × 10−10
𝜀′′
=
tan 𝛿LF = ′
= 0.24
(𝜀r 𝜀0 ) 53.3 × 8.854 187 82 × 10−12
P = 2𝜋(2 × 109 )(1.154 × 10−10 )(0.24)(5)2 = 8.69 W
𝜀′′ =
3 × 108 ∕2 × 109
𝜆
dp = √
=√
√
√
1
1
2𝜋[( 1 + tan2 𝛿LF − 1)] ∕2
2𝜋[( 1 + (.245)2 − 1)] ∕2
= 0.2 m
12.2 RF Dosimetry
RF dosimetry quantifies the magnitude and distribution of electromagnetic
energy absorbed by biological tissue [13]. The specific absorption rate (SAR)
measures the amount of RF energy absorbed by the body. RF dosimetry takes
into account the shape as well as the heterogeneity of the tissues. The unit for
absorbed dose of RF energy (i.e. rate of energy absorption per unit mass) is
W/kg. Some factors that affect dosimetry include [14]:
•
•
•
•
•
•
•
Dielectric constant
Tissue geometry and size
Tissue orientation and field polarization
Field intensity and frequency
Source configuration
Environment
Exposure time.
Estimates of SAR distributions in the body come from measurements in human
models, in animal tissues, or from calculations.
393
394
12 Biological Effects of RF Fields
Table 12.1 Average properties of brain, skull, and muscle tissue [16].
Frequency
100 MHz
800 MHz
1 GHz
2 GHz
5 GHz
𝜺′r (F/m)
𝝈 (S/m)
𝝆 (kg/m3 )
Brain
68.47
0.44
1030.0
Skull
21.45
0.12
1850.0
Muscle
66.19
0.73
1040.0
Brain
46.25
0.73
1030.0
Skull
16.78
0.22
1850.0
Muscle
56.21
0.93
1040.0
Brain
45.43
0.80
1030.0
Tissue
Skull
16.47
0.26
1850.0
Muscle
55.74
1.01
1040.0
Brain
43.21
1.26
1030.0
Skull
15.37
0.48
1850.0
Muscle
54.17
1.51
1040.0
Brain
39.30
3.48
1030.0
Skull
13.05
1.39
1850.0
Muscle
50.13
4.24
1040.0
The National Council on Radiation Protection and Measurements (NCRP)
defines SAR as the time derivative of the incremental energy absorbed by an
incremental mass contained in a volume element of a given density [15]
SAR =
𝜎|Erms |2
𝜎(r)|Erms (r)|2
1
dr ≈
W∕kg
Υ ∫sample
𝜌(r)
𝜌
(12.4)
where
𝜎 = tissue conductivity (S/m)
Erms = RMS electric field
𝜌 = tissue density (kg/m3 )
Υ = volume.
Values of 𝜎 and 𝜌 for brain, skull, and muscle at five frequencies are found in
Table 12.1.
A rise in the tissue temperature due to RF heating contributes to SAR according to [13]
SAR =
cp ΔT
t
W∕kg
where
cp = specific heat (J/g ∘ C)
(12.5)
12.2 RF Dosimetry
ΔT = rise in temperature ( ∘ C)
t = exposure time (seconds).
The specific heat of water is 4.186 J/g ∘ C. An approximate value of cp for bone
is 3.7 J/g ∘ C and for muscle/brain is 1.3 J/g ∘ C [17].
Example
If the rms electric field = 4 V/m, 𝜎 = 150 S/m, 𝜌 = 1250 Kg/m3 , find the SAR and
incident power density.
Solution
Use (12.4) to find SAR = 1.92 W/kg.
Incident power density=
|Erms |2
377 Ω
= 0.042 W/m2 .
Whole-body exposure means that the incident field has a relatively uniform amplitude over the entire biological object [9]. The IEEE standard for
whole-body average specific absorption rate (WBSAR) limits occupational
exposure to 0.4 W/kg and public exposure to 0.08 W/kg [18]. The effect of
body size and shape on WBSAR has been examined for plane wave exposure
[19]. An individual’s height determines the maximum RF energy absorbed at
frequencies with wavelengths on the order of a person’s height. The SAR at
this whole body resonance frequency increases as height decreases [20].
The IEEE standard [21] establishes the whole-body maximum permissible
exposure (MPE) and time averaged exposure limits for electric fields and
magnetic fields (Table 12.2). The IEEE’s MPE limits are spatially averaged
over the whole body under two circumstances: occupational/controlled and
general population/uncontrolled [13]. Occupational/controlled limits apply
to people exposed in a workplace provided; those people know about the
potential for exposure and have the ability to control their exposure. Limits
for occupational/controlled exposure apply when an individual passes through
a location with occupational/controlled limits provided the individual knows
about the potential for exposure. General population/uncontrolled exposures
apply to the general public or workers not fully aware of the potential for
exposure or cannot exercise control over their exposure.
Example
Plot the power density limits in Table 12.2 up to 1000 GHz.
Solution
The data in Table 12.2 was entered into an m-file then graphed as a log–log plot
in Figure 12.6.
395
12 Biological Effects of RF Fields
Table 12.2 Limits for maximum permissible exposure (MPE) [22].
Frequency (MHz)
Power density (W/m2 )
Averaging time (min)
Limits for occupational/controlled exposures
0.1–1.0
9 000
6
1.0–30
9 000/f 2
6
30–300
10
6
f /30
6
5
6
300–3 000
3 000–300 000
Limits for general population/uncontrolled exposure
0.1–1.34
1 000
30
1.34–30
1 800/f 2
30
2.0
30
30–400
400–2 000
2 000–100 000
Power density (W/m2)
396
104
f /200
30
10
30
Occupational/controlled
General population/uncontrolled
102
100 −1
10
100
101
f (GHz)
102
Figure 12.6 Power
density limits for
occupational/controlled
and general
population/uncontrolled
exposures for up to
1000 GHz.
103
12.3 RF Radiation Hazards
After Hertz invented the antenna, people became interested in the biological
effects of RF radiation [23]. In the late 1880s, d’Arsonval investigated the
influence of RF on cells [24]. Researchers knew that shorter-wave RF induced
heating in tissue. Some experiments on humans were done in the 1920s
and 1930s followed by experiments with monkeys [25]. Many years later, a
42-year-old man was working 10 feet in front of a radar antenna [26]. Within
seconds, he felt a sensation of heat that became intolerable in less than a
minute. He moved away from the antenna, and an hour later, he was in a state
of mild shock and died in a little over a week. As a result, safety procedures
12.3 RF Radiation Hazards
were developed for workers in the vicinity of high power radars. This episode
startled people and motivated safer operating procedures around high power
RF. Standards became prevalent and international organizations were formed
to encourage the study of RF effects on humans.
12.3.1
Base Stations
A base station antenna radiates on the order of several tens of watts. The radiated RF fields for rooftops near base stations concern people working or living
near them [20]. A rooftop highly attenuates signals and protects people in a
building from exposure to high field levels.
12.3.2
Cell Phones
Cell phone radiation penetrates approximately 2 cm into the brain at
1800–1900 MHz [27]. For cell phones held against the ear, the SAR drops off
rapidly for the regions of the brain away from the antenna and is negligible for
the rest of the human body except for the hand.
12.3.3
Medical Tests
Magnetic resonance imaging (MRI) potentially causes adverse effects in some
patients, including a potential for uncomfortable exposure to acoustic noise,
heating, and sensory disturbances (in particular vertigo). However, all of these
effects are reversible and can be prevented or ameliorated. Insufficient evidence
prevents drawing firm conclusions about long-term health effects. Theory suggests that no permanent damage results if scanners are operated in line with
existing guidelines that limit the exposure of patients to static and RF fields
during MRI procedures [9]. The RF guidelines avoid excessive elevation of core
body temperature or local temperature in the head, trunk, or extremities by
restricting the SAR.
Healthcare facilities use high power RF for the ablation of tumors and in
diathermy for deep tissue heating [6]. Ablation radiates 200 W at 915 and
2450 MHz and 500 W at 500–750 kHz. Both methods aim to heat the target
tissues to 65–98 ∘ C. Thermal ablation of a number of tumor types, including
liver, breast, thyroid, and prostrate, has a success rate of 95% for a single
treatment, with a three-year overall patient survival rate of 90%. While this
technology is minimally invasive and cause only the local ablation of target
tissues with minimal damage to overlying structures or surrounding tissues,
there are concerns about possible collateral damage to normal structures
adjacent to the desired zone of ablation.
397
398
12 Biological Effects of RF Fields
12.4 Modeling RF Interactions with Humans
Tomographic medical imaging techniques created three-dimensional computer models (called voxel models, tomographic models, or phantoms) based
on the human anatomy [28]. The models match the actual dimensions of
organs.
The National Library of Medicine (NLM) Visible Human Project (VHP) created publicly available complete, anatomically detailed, three-dimensional representations of a human male cadaver and a human female cadaver [29]. The
15 GB Visible Man data set became available in 1994 and the 40 GB Visible
Woman in 1995. The data sets serve as (i) a reference for the study of human
anatomy, (ii) public-domain data for testing medical imaging algorithms, and
(iii) a test bed and model for the construction of network-accessible image
libraries. Figure 12.7 shows a section through the Visible Human Male – head,
including cerebellum, cerebral cortex, brainstem, and nasal passages.
A computational electromagnetics simulation segmented the phantom in
Figure 12.8. A cell phone transmitting 1 W through a PIFA antenna next to the
right ear of the phantom heats the tissue close by. Figure 12.8 shows the SAR
distribution averaged over 10-g of contiguous tissue at 900 MHz induced in
the entire head as well as a plane cut through maximum average SAR [30].
Researchers use specialized laboratory test equipment called Dosimetric
Assessment SYstem (DASY) for conducting SAR measurements. The equipment consists of a “phantom” (human or box), precision robot, RF field sensors,
and mobile phone holder [31]. The phantom contains tissue simulating liquids
that represents the electrical properties of human tissue. The robot moves the
probe through the simulating liquid and measures the SAR vs. position in the
phantom.
Figure 12.7 Male human head from the VHP. Source:
www.nlm.nih.gov.
12.4 Modeling RF Interactions with Humans
Clamp to range: (Min: 0/ Max: 2)
W/kg
Clamp to range: (Min: 0/ Max: 1)
W/kg
2.00
1.00
1.69
0.844
1.44
1.19
0.938
0.719
0.594
0.469
0.688
0.438
0.344
0.219
0.0938
0
0.188
0
SAR (rns)
Type
SAR (f=0.9) [1] (10g) 1W
Monitor
Maximum-3d 4.34433 W/kg at −70.1978 / 133.891 / −65.2669
Frequency 0.9
(a)
Type
Monitor
Plane at y
Maximum-2d
Frequency
SAR (rns)
SAR (f=0.9) [1] (10g) 1W
138.102
4.34428 W/kg at −81.5056 / 138.256 / −63.0326
0.9
(b)
Figure 12.8 The 10-g averaged SAR distributions in the SAM model at 900 MHz: (a) the 3-D
surface SAR distribution and (b) the 2-D SAR distribution in a cutting plane. Source: Zhao
et al. [30]. Reproduced with permission of IEEE.
The change in temperature in a dielectric due to an RF field is expressed as
ΔT =
Pa t ∘
K
mass × cp
(12.6)
where
Pa = power absorbed (W)
t = exposure time (seconds)
mass= mass of object (g)
cp = specific heat (J/(kg ∘ C).
The following procedures measure SAR inside a head phantom [31]:
• Position the handset against the phantom body and switched to full power.
• The precision robot moves the RF probe throughout the phantom head measuring the radio signal level in the head phantom.
• Convert the measured data into SAR (W/kg).
• The full test is conducted at all operating frequencies and using different
phone positions.
• The maximum level measured is recorded as the SAR value against the head.
The following procedures measure SAR inside a body (box) phantom [31]:
• Position the handset against the phantom body and switched to full power.
• The precision robot moves the RF probe throughout the phantom body measuring the radio signal level in the body near the phone.
• Convert the measured data into SAR (W/kg).
• The maximum level measured is recorded as the SAR value against the body.
399
400
12 Biological Effects of RF Fields
12.5 Harmful Effects of RF Radiation
At mobile phone frequencies, most of the energy absorbed by the skin and
other superficial tissues produce a negligible temperature rise in the brain or
any other organs of the body. A person’s ear receives the most RF radiation
from a smart phone. Researchers have found no basis for concerns over auditory perception and on acoustic evoked potentials as well as auditory functions
of the cochlea or auditory brainstem responses [32].
The Surveillance, Epidemiology, and End Results (SEER) Program of the
National Cancer Institute (NCI) tracks US cancer statistics [33]. SEER data
shows even as cell phone use in the United States significantly increased, the
age-adjusted incidence of brain cancer remained unchanged. Wide spread use
of mobile phones did not start until the early 1990s, so epidemiological studies
only assess cancers that occur since then. Animal studies have not shown an
increased cancer risk due to long-term exposure to RF signals.
The International Agency for Research on Cancer (IARC) launched a large
study, Interphone, that looked for links between use of mobile phones and head
and neck cancers in adults [34]. The data from 13 participating countries found
no increased risk of glioma or meningioma (brain tumors) with mobile phone
use of more than 10 years. People who reported the highest 10% of cumulative
hours of cell phone use had some indications of an increased risk of glioma,
although increasing risk with increasing the duration of use did not increase the
risk. No increase in risk of acoustic neuroma (benign tumor on auditory nerves)
was found. The researchers concluded that biases and errors question these
conclusions and prevent a causal interpretation. This study led IARC to classify
RF fields as possibly carcinogenic to humans (Group 2B). In other words, there
is a credible causal association, but chance, bias or confounding cannot be ruled
out with reasonable confidence.
Several studies investigated the impact of RF radiation from cell phones on
brain electrical activity, cognitive function, sleep, heart rate, and blood pressure. No evidence of adverse health effects were found from field levels that
do not cause tissue heating [27]. Some people claim to have electromagnetic
hypersensitivity (EHS) or are sensitive to electromagnetic fields, particularly
radiation from cell phones. The World Health Organization (WHO) concluded
after many studies [35]: “Whatever its cause, EHS can be a disabling problem
for the affected individual. EHS has no clear diagnostic criteria and there is no
scientific basis to link EHS symptoms to electromagnetic field exposure. Further, EHS is not a medical diagnosis, nor is it clear that it represents a single
medical problem.”
Problems
12.1
A 1 GHz plane wave is incident on a dielectric with 𝜀′ = 60𝜀0 and
𝜎 = 1 S/m. Find (a) dielectric loss factor, (b) penetration depth, and (c)
dielectric heating power if the electric field amplitude is 1 V/m.
References
12.2
Plot the penetration depth into the brain for 100 ≤ f ≤ 5000 MHz.
12.3
Plot the penetration depth into the skull for 100 ≤ f ≤ 5000 MHz.
12.4
Plot the penetration depth into the muscle for 100 ≤ f ≤ 5000 MHz.
12.5
Calculate SAR for Table 12.1.
12.6
Plot SAR for brain, muscle, and skull from 100 MHz to 5 GHz when a
5 V/m electric field is present.
12.7
You walk within 10 m of a base station that is radiating 10 W through
an antenna with 10 dB of gain at 2 GHz. Calculate the SAR in the brain.
12.8
How much exposure time is needed to raise the following tissue
1 ∘ C: (a) brain cp = 3.6 J/g ∘ C, (b) skull cp = 2.0 J/g ∘ C, and (c) muscle
cp = 3.4 J/g ∘ C.
12.9
The permittivity and conductivity of many different tissues as a function of frequency are available at https://itis.swiss/virtual-population/
tissue-properties/database/dielectric-properties. Pick a tissue not
listed in Table 12.1. Plot (a) penetration depth vs. frequency and (b)
SAR vs. frequency for a 1 V/m electric field.
References
1 Mahaffey, J. (2015). Atomic Accidents A History of Nuclear Meltdowns and
2
3
4
5
6
7
Disasters; From the Ozark Mountains to Fukushima. New York: Pegasus
Books.
Burda, H., Begall, S., Červený, J. et al. (2009). Extremely low-frequency electromagnetic fields disrupt magnetic alignment of ruminants. Proceedings of
the National Academy of Sciences 106 (14): 5708–5713.
Flores-McLaughlin, J., Runnels, J., and Gaza, R. (2017). Overview of
non-ionizing radiation safety operations on the International Space Station.
Journal of Space Safety Engineering 4 (2): 61–63.
http://physicscentral.com/explore/action/radiationandhumans.cfm (17
November 2018).
https://www.fda.gov/radiation-emittingproducts/
radiationemittingproductsandprocedures/homebusinessandentertainment/
cellphones/ucm116282.htm (accessed 17 November 2018).
(2006). Framework for Developing Health-Based EMF Standards. Switzerland: World Health Organization Press, World Health Organization Press.
Cleveland, R.F. Jr. and Ulcek, J.L. (1999). Questions and Answers About
Biological Effects and Potential Hazards of Radiofrequency Electromagnetic
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9
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13
14
15
16
17
18
19
20
21
22
23
Fields, 4e. Federal Communications Commission Office of Engineering &
Technology, OET Bulletin 56.
Baker-Jarvis, J. and Kim, S. (2012). The interaction of radio-frequency fields
with dielectric materials at macroscopic to mesoscopic scales. Journal of
Research of the National Institute of Standards and Technology 117: 1–60.
Chou, C.K., Bassen, H., Osepchuk, J. et al. (1996). Radio frequency
electromagnetic exposure: tutorial review on experimental dosimetry.
Bioelectromagnetics 17 (3): 195–208.
https://www.microdenshi.co.jp/en/microwave (accessed 17 November 2018).
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November 2018).
Johnk, C.T.A. (1988). Engineering Electromagnetic Fields and Waves, 2e.
New York: Wiley.
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http://www.who.int/peh-emf/meetings/04_Chou.pdf (accessed 19 November
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https://www.everythingrf.com/rf-calculators/sar-rf-exposure-calculator
(accessed 11 December 2018).
https://www.fcc.gov/general/body-tissue-dielectric-parameters (11 December
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Giering, K., Lamprecht, I., and Minet, O. (1996). Specific heat capacities of
human and animal tissues. In: SPIE Proceedings Volume 2624, Laser-Tissue
Interaction and Tissue Optics, SPIE = The International Society of Optics
and Photonics, 188–197.
IEEE Standard C95.1 2005. IEEE Standard for Safety Levels with Respect
to Human Exposure to RadioFrequency Electromagnetic Fields, 3 kHz to
300 GHz, IEEE.
Hirata, A., Fujiwara, O., Nagaoka, T., and Watanabe, S. (2010). Estimation
of whole-body average SAR in human models due to plane-wave exposure
at resonance frequency. IEEE Transactions on Electromagnetic Compatibility
52 (1): 41–48.
Gandhi, O.P. (1929). Dosimetry—the absorption properties of man and
experimental animals. Bulletin of the New York Academy of Medicine 55
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research on the biological effects of microwave radiation: 1940–1960.
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References
24 d’Arsonval, A. (1893). ’Influence de l’é1ectricité sur la cellule microbienné.
Arch. de physiol, norm. et path. 5 (5): 66–69.
25 Turner, J.J. (1962). The Effects of Radar on the Human Body. Whippany, NJ,
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McLaughlin, J.T. (1957). Tissue destruction and death from microwave radiation (radar). California Medicine 86 (5): 336–339.
National Research Council (2008). Identification of Research Needs Relating
to Potential Biological or Adverse Health Effects of Wireless Communication
Devices. Washington, DC: The National Academies Press.
Caon, M. (2004). Voxel-based computational models of real human
anatomy: a review. Radiation and Environmental Biophysics 42 (4):
229–235.
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November 2018).
Zhao, L., Ye, Q., Wu, K. et al. (2016). A new high-resolution electromagnetic human head model: a useful resource for a new
specific-absorption-rate assessment model. IEEE Antennas and Propagation
Magazine 58 (5): 32–42.
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Sievert, U., Eggert, S., and Pau, H.W. (Apr 2005). Can mobile phone emissions affect auditory functions of cochlea or brain stem? Otolaryngology
Head and Neck Surgery 132 (3): 451–455.
http://seer.cancer.gov (accessed 23 October 2018).
The INTERPHONE Study Group (2010). Brain tumour risk in relation
to mobile telephone use: results of the INTERPHONE international
case–control study. International Journal of Epidemiology 39 (3): 675–694.
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403
405
Appendix A
MATLAB Tips
A.1 Introduction
If you are unfamiliar with MATLAB or need to refresh your skills, I suggest
starting at https://www.mathworks.com/support/learn-with-matlab-tutorials
.html
A search of the Internet will produce many books and helpful resources on
MATLAB.
There are a wide range of MATLAB toolboxes dedicated to topics in wireless
communications. I tried to stick to basic MATLAB in my examples and homework problems, so that readers do not have to spend a lot of money buying
toolboxes. If you want to become a wireless communications MATLAB guru,
then these are the most relevant MATLAB packages:
MATLAB and SIMULINK
Signal Processing Toolbox
RF Toolbox
Phased Array Toolbox
Communications Toolbox
DSP System Toolbox
LTE Toolbox
Audio Toolbox
WLAN Toolbox
Antenna Toolbox
5G Toolbox
If you do not have money to buy MATLAB, then here are some cheap alternatives:
• There are free online calculators for many of the problems in this book.
• Use Python – it is MATLAB – like and is free.
• Other programming languages and math software tools.
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
406
Appendix A MATLAB Tips
A.2 Plotting Hint
My pet peeve with MATLAB is the default graphics. Everything is too
tiny – font and line widths. I suggest that you create an m-file called startup
and put it in the bin directory in MATLAB. Everytime you start MATLAB,
these commands are executed. You can put additional commands in this file.
Here are some helpful commands for plotting:
set(0,'DefaultAxesFontSize',28)
set(0,'DefaultLineLineWidth',2)
set(0,'DefaultTextFontSize',28)
set(0,'DefaultAxesLineStyleOrder','-|--|:|-.')
set(0,'DefaultAxesColorOrder',[0 0 1;1 0 0;0 0 0;0 1 0])
set(0,'DefaultAxesFontName','Times New Roman')
set(0,'DefaultTextFontName','Times New Roman')
These commands save you from changing these parameters every time you call
MATLAB. Your graphics will look much better!
407
Appendix B
OSI Layers
In 1983, the International Standards Organization (ISO) developed the Open
Systems Interconnection (OSI) model to define a framework for computer
communications (Figure B.1). The OSI model has seven layers described below
in detail. Lower layers (1–4) concern moving data around, while upper layers
(5–7) deal with application-level data. A mnemonic trick for memorizing the
seven layers: “Please Do Not Tell Secret Passwords Anytime.”
B.1 Layer 1: Physical
The physical layer transmits signals across a communication medium. It
includes but is not limited to cables; antennas; electronics; power; bit rate;
point-to-point; multipoint or point-to-multipoint line configuration; network
topology; serial or parallel communication; simplex, half duplex or full duplex
transmission mode; modulation; line coding; synchronization; circuit switching; multiplexing; equalization; training sequences; pulse shaping; FEC; and
bit interleaving.
B.2 Layer 2: Data Link
The data link layer transforms bits from the physical layer into a frame for
the network layer. It provides direct node-to-node data transfer, and error
correction from the physical layer. The Data Link Layer has encoding and data
compression. Two sublayers in the data link are the Media Access Control
(MAC) layer and the Logical Link Control (LLC) layer. In the networking
world, most switches operate in layer 2.
B.3 Layer 3: Network
The network layer controls packet routing and works with IP addresses.
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
408
Appendix B OSI Layers
Figure B.1 OSI layers.
Transmit
data
Users
Receive
data
Application
Presentation
Session
Transport
Network
Data link
Physical
Physical link
B.4 Layer 4: Transport
The transport layer splits the data from the session layer into packets for delivery on the network layer and verifies that the packets arrive error free at the
other end. It decides how much data to send, the data rate, destination, etc. The
Transmission Control Protocol (TCP) occurs at this layer.
B.5 Layer 5: Session
The session layer establishes and manages sessions, conversions, or dialogues
between two computers. Two devices communicate via a session. Functions
at this layer involve setup, coordination, and termination between the applications at each end of the session. Examples include: IPv4, IPv6, and Apple Talk.
B.6 Layer 6: Presentation
The presentation layer manages the syntax and semantics of the information
transmitted between two computers. It represents the preparation or translation of application format to network format, or from network formatting to
application format. This layer prepares data for the application or the network.
Encryption and decryption happens in this layer. Examples include ASCII,
JPEG, tif, and gif.
Appendix B OSI Layers
B.7 Layer 7: Application
The application layer deals with data symbols and has several common protocols, such as file transfer, virtual terminal, and email. Most users only deal with
this layer. Some examples include browsers, email, video conferencing, HTTP
(HyperText Transfer Protocol), and FTP (File Transfer Protocol).
409
411
Appendix C
Cellular Generations
First generation (1G) wireless systems were analog systems that only supported voice calls. Nippon Telephone and Telegraph (NTT) in Tokyo, Japan,
Europe Nordic Mobile Telephone (NMT), and Bell labs AMPS (Advanced
Mobile Phone Service) started service in the early 1980s. The desire for data
communication, security, and high data rates initiated a digital trend with the
second generation (2G) and continues today [1–3]. Table C.1 compares the
four generations of cell phones.
Table C.1 Comparison of wireless generations [4].
Generation
1
2
3
4
Introduced
1981
Introduced
USA
1991
2000–2002
2009
Finland
Japan
South Korea
Technology
AMPS, NMT,
TACS
IS-95, GSM
IMT2000,
WCDMA
LTE, WiMAX
Multiple Access
FDMA
TDMA, CDMA CDMA
Switching type
Circuit
switching
Packet switching Packet switching
Circuit
except for Air
switching for
Interface
Voice and
Packet switching
for Data
Data rate
2.4–14.4 kbps
14.4 Kbps
3.1 Mbps
Supports
Voice only
Voice and Data
Voice and Data
Voice and Data
Internet
None
Narrowband
Broadband
Ultra Broadband
Bandwidth
Analog
25 MHz
25 MHz
100 MHz
CDMA
100 Mbps
(Continued)
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
412
Appendix C Cellular Generations
Table C.1 (Continued)
Generation
1
2
3
4
Operating
frequencies
800 MHz
GSM: 900,
1800 MHz
CDMA:
800 MHz
2100 MHz
850, 1800 MHz
Advantage
Simpler (less
complex)
network
elements
Multimedia
features (SMS,
MMS), Internet
access and SIM
introduced
High security,
international
roaming
Speed, high
speed handoffs,
MIMO
technology,
Global mobility
Disadvantages
Low network
Limited
range, slow data
capacity, not
rates
secure, poor
battery life, large
phone size,
background
interference
High power
consumption,
Low network
coverage, High
cost of spectrum
licence
Hard to
implement,
complicated
hardware
required
Applications
Voice Calls
Voice calls,
Video
High speed
Short messages, conferencing,
applications,
browsing
mobile TV, GPS mobile TV,
Wearable
devices
References
1 Vora, L.J. (2015). Evolution of mobile generation technology: 1G to 5G and
review of upcoming wireless technology 5G. International Journal of Modern
Trends in Engineering and Research 2 (10): 281–290.
2 Bhandari, N., Devra, S., and Singh, K. (2017). Evolution of cellular network:
from 1G to 5G. International Journal of Engineering and Techniques 3 (5):
98–105.
3 Mohammad Meraj ud in Mir , Dr. Sumit Kumar (2015). Evolution of mobile
wireless technology from 0G to 5G. International Journal of Computer
Science and Information Technologies 6 (3): 2545–2551.
4 http://www.zseries.in/telecom%20lab/telecom%20generations/#.XEE83lxKhPY
(accessed 17 January 2019).
413
Appendix D
Bluetooth
In 1994, Ericsson invented a short distance wireless technology for exchanging data between fixed and mobile devices over the ISM (Industrial, Scientific,
and Medical) band 2.4–2.4835 GHz [1]. It was named after the Scandinavian
king Harald Bluetooth. Bluetooth ( ) operates at frequencies between 2402
and 2480 MHz, or 2400 and 2483.5 MHz including guard bands 2 MHz wide
at the bottom end and 3.5 MHz wide at the top as shown in Figure D.1 [2].
It uses frequency-hopping spread spectrum (FHSS) at 1600 hops per second.
The data packets transmit on one of 79 designated channels. Each channel has
B = 1 MHz with 800 hops per second. Bluetooth has many updates starting with
version 1 and extending to the current version 5.
Bluetooth Low Energy (BLE) occupies the same frequency band as Bluetooth
using 40 B = 2-MHz channels and operates with significantly lower power consumption and cost. BLE enables small sensors to operate off tiny batteries for
months.
The original Bluetooth used Gaussian frequency-shift keying (GFSK)
modulation. Devices using GFSK operate in basic rate (BR) mode at 1 Mbit/s.
Enhanced Data Rate (EDR) mode came later with 𝜋/4-DPSK modulation at
2 Mbit/s and 8-DPSK modulation at 3 Mbit/s. Bluetooth that combined the BR
and EDR modes is called BR/EDR radio.
Bluetooth has a master/slave architecture for passing packets. One master
communicates with up to seven slaves that synchronize to the master’s clock
that has a period of 312.5 μs. Two clock periods make up a slot of 625 μs, and
two slots make up a slot pair of 1250 μs. The master transmits in even slots
and receives in odd slots for single-slot packets. The slave receives in even slots
and transmits in odd slots. Packets may be 1, 3, or 5 slots long. The master
rapidly switches from one device to another checking for a slave to address.
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
Appendix D Bluetooth
Figure D.1 Bluetooth spectrum.
2.4 GHz
Guard band
Hopping channels
Guard band
414
…
B
2.4835 GHz
References
1 Morrow, R. (2002). Bluetooth: Operation and Use. New York: McGraw-Hill.
2 https://www.edgefxkits.com/blog/bluetooth-technology-and-its-working
(accessed 3 February 2019).
415
Appendix E
Wi-Fi
Wi-Fi represents the IEEE 802.11* standards where * is a, b, e, f, g, h, I, j, k, n,
s, u, ac, ad, af, ah, and ax [1]. A worldwide network of companies called the
Wi-Fi Alliance owns the Wi-Fi registered trademark. Wi-Fi provides broadband Internet connection via access points over an area known as a hotspot.
The 802.11 standards specify Wi-Fi communications at bands around 900 MHz,
2.4 GHz, 3.6 GHz, 4.9 GHz, 5 GHz, 5.9 GHz, and 60 GHz bands. Each frequency
band has many channels that countries regulate in terms of allowable channels, allowed users, and maximum power levels within these frequency ranges.
Bandwidth requirements increased with the evolution of 802.11 standards.
Wi-Fi Alliance started referring to Wi-Fi by number versions rather than
using the very confusing IEEE standard designation that has letters that make
no sense to most people. All new versions of Wi-Fi receive one higher number
than the previous version [2]. Higher number are compatible with lower numbers and have higher performance. This new designation does not apply to the
outdated Wi-Fi 1–3.
Table E.1 summarizes the characteristics of the different Wi-Fi releases. In
the 2.4-GHz ISM band microwave ovens, cordless telephones, USB 3.0 hubs,
and Bluetooth devices interfere with Wi-Fi. The United States has 11 channels
in the 2.4 GHz band, while Australia and Europe have 13, and Japan 14. A Wi-Fi
signal only occupies two or three channels spaced across the 2.4 GHz band in
order to limit interference between channels.
CCK (complementary code keying) is a spread spectrum technique for low
data rates up to 11 Mbps. 802.11a/n/ac use the 5 GHz U-NII band, which, for
much of the world, offers at least 23 nonoverlapping 20 MHz channels rather
than the 2.4 GHz ISM frequency band, where the channels are only 5 MHz
wide. Common building materials absorb frequencies near 5 GHz, resulting in
a shorter range.
802.11n uses double the radio spectrum/bandwidth (40 MHz) compared to
802.11a or 802.11g (20 MHz). This means there can be only one 802.11n network on the 2.4 GHz band at a given location, without interference to/from
other WLAN traffic. 802.11n can also be set to limit itself to 20 MHz bandwidth
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
416
Appendix E Wi-Fi
Table E.1 Characteristics of Wi-Fi 1–Wi-Fi 6 [2, 3].
Max.
rate
Frequency
(GHz)
Channel
width
(MHz)
Modulation
Multiplexing
Generation
Standard
Date
1
802.11b
1999 11 Mbps
2.4
20
CCK
2
802.11a
1999 54 Mbps
5
20
BPSK, QPSK, OFDM
16-QAM,
64-QAM
3
802.11g
2003 54 Mbps
2.4
20
CCK, DSSS
OFDM
4
802.11n
2009 600 Mbps
2.4 or 5
20 or 40
CCK, DSSS
OFDM
5
802.11ac
2014 6.93 Gbps 5.8
20, 40,
and 80
BPSK, QPSK, OFDM
16-QAM,
64-QAM,
256-QAM
6
802.11ax 2019 14 Gbps
20, 40, 80,
and 160
BPSK, QPSK, OFDMA
16-QAM,
64-QAM,
256-QAM,
1024-QAM
2.4 and 5
to prevent interference in dense community. Wi-Fi 4 was first to use multiple
input/multiple output (MIMO).
Salient features of Wi-Fi 6 include [4]:
• Uplink and downlink orthogonal frequency division multiple access
(OFDMA) increases efficiency and lowers latency for high demand
environments
• Multi-user multiple input, multiple output (MU-MIMO) allows more data
to be transferred at one time, enabling access points (APs) to handle larger
numbers of devices simultaneously
• Transmit beamforming enables higher data rates at a given range to increase
network capacity
• quadrature amplitude modulation mode (1024-QAM) increases throughput
for emerging, bandwidth-intensive use cases
• Frequency Division Multiple Access or OFDMA. The Wi-Fi access point can
talk to more devices at once.
References
1 https://www.electronics-notes.com/articles/connectivity/wifi-ieee-802-11/
standards.php (accessed 17 January 2019).
Appendix E Wi-Fi
2 Kastrenakes, J. (2018). Wi-Fi now has version numbers, and Wi-Fi 6 comes
out next year. https://www.theverge.com/2018/10/3/17926212/wifi-6-versionnumbers-announced (accessed 10 March 2018).
3 https://ccm.net/contents/802-introduction-to-wi-fi-802-11-or-wifi (accessed
14 February 2019).
4 IEEE 802.11ax: The Sixth Generation of Wi-Fi, Cisco Technical White Paper.
https://www.cisco.com/c/dam/en/us/products/collateral/wireless/white-paperc11-740788.pdf (accessed 14 February 2019).
417
419
Appendix F
Software-Defined Radios
Radios existed long before computers and software, so the idea of replacing
some of the hardware in a radio with a computer and software did not occur
until late in the twentieth century. Software-defined radio (SDR) alters the
transmit/receive waveform in software rather than hardware. Traditional
hardware functions, such as modulation, carrier frequency, or coding, now
become lines of code.
F.1
SDR Basics
The benefits of SDR are:
• Flexibility: Requires no hardware changes when switching between functions
• Interoperability: Works with old and new systems that have appropriate software
• Ease of upgrade: Adds new features and advances through software updates
• Efficiency: Supports many different radios
• Higher-level interfaces: Operates through GUI and network interfaces.
The ultimate goal of SDR technology is to have a radio that communicates at
any frequency, bandwidth, modulation, and data rate through software updates
rather than hardware changes. The ideal SDR is [1]
1. Multiband: Operates on two or more bands either sequentially or simultaneously.
2. Multicarrier or multichannel: Simultaneously operates at more than one frequency within the same band or in two different bands at the same time.
3. Multimode: The ability to switch modulation schemes.
4. Multirate: The ability to process different data rates.
5. Variable bandwidth: Channel bandwidth determined by digital filters.
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
420
Appendix F Software-Defined Radios
Antenna
BPF
PA
Mixer
BPF
Amp
Mod
DAC
Computer
ADC
Computer
LO
LO
RF stage
IF stage
(a)
Antenna
BPF
LNA Mixer
BPF
LO
Amp Demod
IF stage
RF stage
(b)
Figure F.1 Block diagram of a two-stage hardware radio transmitter and receiver.
(a) Transmitter and (b) receiver
The SDR Wireless Innovation Forum (WINNF) defines five tiers of radios based
on which parts are configurable [2].
Tier 0: All functions in Figure F.1 are hardware. No functions are software
reconfigurable.
Tier 1: A radio with limited functions controlled by software, such as power
levels and interconnections, but not mode or frequency. All functions in
Figure F.1 are performed in hardware with some software control.
Tier 2: A radio with a hardware-based RF front end but software control of frequency, modulation and waveform generation/detection, wide/narrowband
operation, and security (Figure F.2).
Tier 3: Nearly all functions performed in software. The analog to digital convertors (ADCs) and digital-to-analog converters (DACs) are as close to the
antenna as possible (Figure F.3).
Tier 4: Full programmability and supports a broad range of functions and frequencies. With many electronic items such as cellphones having many different radios and standards a software definable multifunction phone would
fall into this category (Figure F.3).
Appendix F Software-Defined Radios
Figure F.2 An IF SDR
transmitter and receiver
eliminates the IF stage. (a)
Transmitter and (b) receiver
Antenna
LPF
PA
RF stage
Mixer
DAC
Computer
ADC
Computer
LO
(a)
Antenna
BPF
LNA
RF stage
Mixer
LO
(b)
Antenna
BPF
Antenna
PA
DAC
RF stage
BPF
LNA
Computer
ADC
Computer
RF stage
(a)
(b)
Figure F.3 SDR DUC transmitter and DDC receiver. (a) Transmitter and (b) receiver
F.2
SDR Hardware
Figure F.1 is a simplified example of a typical two-stage transmitter and receiver.
The radio frequency (RF) and IF (intermediate frequency) stages are hardware.
The DAC in the transmitter (Figure F.1a) sends an analog baseband signal to the
IF stage. The IF stage upconverts the baseband frequency to an intermediate
frequency. This signal is amplified then passed through a bandpass filter (BPF).
The IF stage output goes to the RF stage input where it is upconverted to RF,
amplified, and filtered before heading to the antenna. The receiver has a reverse
process in which the receive antenna sends its signal to the RF stage where it is
filtered, amplified, and downconverted to an IF (Figure F.1b). The IF stage then
421
422
Appendix F Software-Defined Radios
filters, amplifies, and demodulates the IF signal. An ADC at the output of the
IF stage converts the IF output to binary format for input to a computer. The
RF and IF stages are designed to interface the ADC input and DAC output with
the antenna.
The antenna and ADC specifications restrict SDR performance [1]. Most
antennas have a narrow bandwidth that limits multiband operation. Systems
operating over a wide bandwidth require a matching circuit that adapts to the
operating frequency changes. The antenna gain also changes with frequency,
so the antenna must adapt its size as the frequency changes. The ADC and
DAC have an upper frequency limit. Signals above that limit require frequency
downconvertion or upconversion.
Figure F.2a shows a transmitter that inputs the bits from the DAC into an RF
stage without need for the IF stage. The RF stage is necessary, because the DAC
output frequency is too low. Once the IF signal is up converted, then it passes
through a power amplifier and BPF before going to the antenna. Figure F.2b is
the receiver without an IF stage.
A direct upconversion (DUC) transmitter (Figure F.3a) passes digital data
through a power amplifier and BPF without the need for upconversion. A direct
downconversion (DDC) receiver (Figure F.3b) passes the RF signal through a
BPF and low noise amplifier (LNA) before the input of the ADC without the
need of an IF stage.
The mass-produced DVB-T TV tuner dongle based on the RTL2832U
chipset serves as the basis for the RTL-SDR [3]. The RTL maximum sample
rate is 3.2 MS/s (mega samples per second). This rate drops samples, though, so
2.4 MS/s is more realistic [4]. The ENOB is 7 bits. These dongles were intended
for TV, so they have a 75 Ω input impedance. Newer dongles have 50 Ω SMA
connectors. Figure F.4 is an example of an RTL SDR with a monopole antenna
plugged into the USB port of a laptop computer.
F.3
SDR Software
The Software Communications Architecture (SCA) specifies how the hardware
and software in an SDR interface with each other. It uses CORBA (Common
Object Request Broker Architecture) which enables software components written in multiple computer languages and running on multiple computers to work
together. The advantages of SCA are [2]:
• Software modules written by different sources are compatible.
• Reusing modules significantly reduce costs.
For amateurs, GNU radio serves the same purpose as SCA and is supported
by a community of developers [5]. GNU radio is free and distributed under the
terms of the GNU General Public Licence (GPL). It can be used with readily
Appendix F Software-Defined Radios
Figure F.4 An RTL SDR.
available low-cost external RF hardware to create software-defined radios, or
without hardware in a simulation-like environment. All of the code is copyright
of the Free Software Foundation.
MATLAB serves as another avenue for interfacing with RTL-SDR [6].
F.4
Cognitive Radio
Cognitive radio is an SDR capable of altering its operating behavior based on
an awareness of its environment. The benefits of CR include [7]
• Optimize use of spectrum
• Organize interoperability
• Map locations of units, rank candidates for dispatch (nearest, equipment and
training, ready to move)
• Reconfigure networks to meet current needs
• Respond to priority structures
• Reach hidden nodes.
References
1 Fette, B.A. (2007). Basics of software defined radio, part 1. EE Times.
2 Mitola, J. III. Software radios: Survey, critical evaluation and future direc-
tions. IEEE Aerospace and Electronic Systems Magazine 8 (4): 25–36.
3 Laufer, C. The Hobbyist’s Guide to the RTL-SDR, 4e.
4 https://www.rtl-sdr.com/about-rtl-sdr (accessed 17 January 2019).
423
424
Appendix F Software-Defined Radios
5 https://www.gnuradio.org/about (accessed 9 January 2019).
6 Stewart, R.W., Barlee, K.W., Atkinson, D.S.W., and Crockett, L.H. (2017).
Software Defined Radio Using MATLAB & Simulink and the RTL-SDR.
Glasgow, Scotland: University of Strathclyde.
7 Cook, P.G. (2007). Introduction to software defined radio, cognitive radio
and the SDR Forum. http://www.npstc.org/download.jsp?tableId=37&
column=217&id=1280&file=IO-Cook-0802145%20NPSTCa.pdf (accessed
25 June 2019).
425
Index
a
access point (AP) 373–374, 381, 382
acknowledgement (ACK) 39
adaptive antenna 325–341
adaptive equalizer 61
Adcock array 308–310
additive white Gaussian noise (AWGN) 32,
302
advanced encryption standard (AES) 382
ALOHA 97–99, 288
Alouette 261
Amateur Satellite Corporation (AMSAT) 243
ambient temperature 30
amplifier 30–32, 83–84, 255, 422
cascade 31–32
amplitude modulation (AM) 72–78
amplitude modulation index 75
amplitude modulation spectrum 76
amplitude shift keying (ASK) 78–80, 278, 280
analog to digital convertor (ADC) 36–38,
144, 422
angle of arrival (AOA) 301–322
antenna array 130–145
antenna beamwidth 133
antenna gain 112–113
antenna impedance 111, 285, 290
antenna pattern 112
antenna temperature 257
anti-collision protocol 288
aperture antenna 125–128
aperture efficiency 113
array factor 131–133, 137, 305
array steering matrix 302
atmospheric attenuation 225–226, 254, 257
oxygen 225–226
water vapor 225–226
atmospheric density 187
atmospheric duct 226–227
atmospheric index of refraction 187
atmospheric refraction 188
attack surface 374
authentication 376, 380, 381
autoID 267, 270
automatic repeat request (ARQ) 39
average fade duration (AFD) 224
axial ratio (AR) 114–115, 124
b
backscatter 275, 285, 288–293
bandwidth
antenna 115
signal 27
system 27
bar code 267–268
base station 151, 397
basic service area (BSA) 374
basic service set (BSS) 373–374
battery-assisted tag (BAT) 271, 277
baud rate 27
beacon 278–279
beam cancellation 327–328
beam switching 340
beamwidth 133, 135, 305
bent pipe architecture 259
binary frequency shift keying (BFSK) 83–84
binary phase shift keying (BPSK) 85–86
bipolar return to zero 27
bit error rate (BER) 33, 94
blackbody equivalent temperature 258
blind equalizer 61
block cipher 379
block codes 45–47
Bluetooth 100, 371, 413
breakpoint 183
Wireless Communications Systems: An Introduction, First Edition. Randy L. Haupt.
© 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.
426
Index
Brewster’s angle 177–178
Brussels International Sunspot Number 230
Bullington method 195–196
burst errors 33
Butler matrix 145, 340
c
Caesar cipher 376
camouflage antenna 155–156
cancellation beam 327–328
Capon’s minimum variance 313–315
carrier 71
carrier-to-noise ratio (CNR) 29
Carson’s rule
FM 82
PM 84
Cassini 255–256
cell 151–154, 261
cell reuse 152–153
cell size 153–154
cell splitting 153
cellular generation 411
cellular network 369
channel 173
channel capacity 100
channel coding 39–40
channel impulse response function 205
channel matrix 349–352
channel sounding 348, 353
channel state information (CSI) 352
channel state information receive (CSIR)
352, 360–361
channel state information transmit (CSIT)
352, 360–361
channel transfer function 54, 204–205, 348
characteristic impedance 7–8
charge pump 275, 277
check digit 268
chip 103
cipher block chaining (CBC) 379
cipher feedback (CFB) 379
ciphertext 378–379
circular aperture 126
circular array 305
circular polarization 114, 184, 285
client-to-client attack 375
codebook 22
code division multiple access (CDMA)
104–105, 253
code division multiplexing (CDM) 104
code rate 40
codeword 25, 44–45
cognitive radio 423
coherence time 166, 205, 222
collision 380
Common Object Request Broker Architecture
(CORBA) 422
confidence level (CL) 34
confidentiality 381
conformal array 135
constellation 85–92
constraint length 47
convolutional codes 47–48
Cornu spiral 191
correlation 54
counter (CTR) mode 379
covariance matrix 61, 303, 310, 313, 351
critical angle 177–178
critical frequency 233
cross-polarization 128, 189, 227
cross-polarization discrimination (XPD) 227
cryptography 376–381
CSI at the receiver (CSIR) 352, 355–356, 360
CSI at the transmitter (CSIT) 352, 355–356,
360
cumulative density function (CDF) 35–36,
208–209
cyber–physical system 369–370
cyclic redundancy check (CRC) 43–45
d
decision-feedback equalizer (DFE) 62
decorrelation distance 216
decription 378
Deep Space Network (DSN) 255–256
demultiplexing 95
denial of service (DoS) 375
dense reader environment 287–288
depolarization 227
Dickson multiplier 276
differential phase shift keying (DPSK) 90
diffraction 190–202
diffraction angle 201
diffraction coefficient 200–201
diffraction from a wedge 200–202
diffraction from multiple obstacles 194–198
diffraction loss 191–192
digital beamforming 144–145
digital to analog converter (DAC) 144, 422
dipole antenna 117–118, 185, 271
Direct Broadcast Satellite (DBS) 241
direct downconversion (DDC) 422
Index
direction finding (DF) 301–322
direction of arrival (DOA) 301
directivity 112–113, 125, 126, 133, 135
direct matrix inversion (DMI) 332
direct sequence spread spectrum (DSSS) 103
direct upconversion (DUC) 422
dispersion 52, 58, 179–180
dispersive channel 51
diversity 162–166, 345
frequency 165
gain 346
polarization 165–166
selection 163
spatial 162–163, 346
time 166
Doppler shift 219–221
Doppler spectrum 221, 223
Doppler spread 204–205, 219, 222
Dosimetric Assessment System (DASY) 398
double sideband (DSB) 72–73
downlink (DL) 242, 243, 250, 255, 259, 260,
261
duplex 94
e
earth curvature 186
Eb/N0 33
Echo 241
effective number of bits (ENOB) 37
EGC diversity gain 164
eigenbeam 310–313
electromagnetic compatibility (EMC) 12
electromagnetic hypersensitivity (EHS) 400
electromagnetic interference (EMI) 12, 33
electron density 233
electronic beam steering 136
electronic codebook (ECB) 379
electronic product code (EPC) 270
electrostatic discharge (ESD) 12
element pattern 137
encription 378
end-fire array 136
energy 28–29, 33, 93–94, 393–394
energy harvesting 274–277
Enigma cipher 377
entropy 6
entropy coding 22, 23
envelope 74–76
envelope detector 74–75
EPCglobal 270
Epstein–Peterson model 197–198
equal gain combining (EGC) 164
equalization 57–62
equivalent noise temperature 31
Error Correcting Codes (ECC) 45–48
estimated planetary K index 231
Estimation of Signal Parameters via Rotational
Invariance Techniques (ESPRIT)
319–321
EUI filtering 382
extended service area (ESA) 374
extended service set (ESS) 373–374
extended unique identifier (EUI) 366
eye-diagram 50–51, 85–86
f
fade margin 202
fading 179, 202–219
channel 350
Doppler 220
fast 205
flat 204
frequency-selective 204
large-scale 203–204, 216–217
Rayleigh 205–209, 223
Rician 209–212
slow 204–205, 221–222
small-scale 203, 205–212
solar 248
far field 116, 123
fast frequency hopping (FFH) 102
finite impulse response filter 59
fixed length code 23
Fixed Satellite Service (FSS) 241
FM0 280, 283
folded dipole 119, 120, 143
forward error correcting (FEC) codes 48
Fourier transform 18, 125, 145
frame 21, 253
free space 7
frequency division duplexing (FDD) 94–95
frequency division multiple access (FDMA)
99, 261
frequency division multiplexing (FDM)
95–96
frequency hopping spread spectrum
100–102, 287
frequency modulation (FM) 80–84
frequency modulation index 80–82
frequency modulation radio 82
frequency modulation spectrum 81–82
Frequency of Optimum Traffic (FOT) 233
427
428
Index
frequency shift keying (FSK) 83–84, 280,
283, 284
Fresnel attenuation 196
Fresnel diffraction 190–197
Fresnel integral 190–191
Fresnel reflection coefficient 176, 189
Fresnel transmission coefficient 176–177
Fresnel zone 192–194, 291–292
Fresnel zone ellipse 193–194
Friis transmission formula 174, 181, 255, 257,
277, 278, 290
full duplex (FDX) 94, 272–273
g
Gaussian minimum shift keying (GMSK)
89–90
generating polynomial 43, 47–48
generator matrix 45
geomagnetic index 231
geometrical optics (GO) 173, 217
geometrical theory of diffraction (GTD)
198–202
geostationary satellite 249
Global Navigation Satellite System (GNSS)
230
Global Positioning System (GPS) 230,
250–254
global search algorithm 340
Global System for Mobile (GSM) 146
GNU radio 422
go-back-N-ARQ 40
Gold codes 104, 253
ground permittivity 184–185
ground reflection coefficient 181–182,
184–185
G/T 257–258
Guglielmo Marconi 3
h
Hadamard matrix 104
half duplex (HDX) 95, 272–273
Hamming distance 25, 45
Hamming weight 25, 41
handoff 152
handover (HOW) 254
handset antenna 146–151
hash 381
hash function 378, 380–381
hashing 378, 380–381
hash table 380–381
Hata attenuation 212–213
helical antenna 124, 146–147, 243, 253
Hertz, Heinrich 2–3
history of wireless 1–4
Hogg antenna 160, 241, 243
i
IEEE 802-11 382, 415–416
implanted medical device 371–372
independent basic service set (IBSS) 373–374
industrial, scientific, and medical (ISM) 11,
202
infinite impulse response (IIR) 57–58
information 4
inlay 270
Inmarsat 249
insertion attack 375
integrity 376, 381
interception attack 375
interference 101, 325–326
interleaving 48–50
International Standards Organization (ISO)
272, 407
International Telecommunication Union (ITU)
11
internet of things (IoT) 369
internet protocol (IP) 21, 366–368, 370
internet protocol version 4 (IPv4) 367
interrogation zone (IZ) 270, 274, 285, 287
Intersymbol Interference (ISI) 17, 51–54, 57,
91
intrusion detection system (IDS) 383
inventory round 270
ionizing radiation 389
ionosonde 233
ionosphere 227–234
D layer 228
E layer 228–229
F layer 230
sporadic E 229
spread F 230
Iridium 249–250
j
jamming attack
jitter 51, 57
375
k
key 378
private 379
public 380
kill command 273
Index
l
laser direct structuring (LDS) 149
latency 249, 371
leaky feeder 127, 156
least mean square (LMS) 329–332
level crossing rate (LCR) 223–224
linear array 131
line of sight (LOS) 52, 181, 192
link budget 175, 254–258, 278, 289
listen before talk 287
local area network (LAN) 365
locally administered address (LAA) 366
log periodic dipole antenna (LPDA) 121
longitudinal redundancy check (LRC) 42–43
loop antenna 122–124, 271, 281, 307
lossless coding 22
lossy coding 22
lossy dielectric 392
low earth orbit satellite 249–250
lowest usable frequency (LUF) 233
low noise amplifier (LNA) 32
low sidelobe taper 138–140, 326
Bayliss 308
binomial 138–140
Dolph–Chebyshev 138–140
Taylor 138–140, 308
m
magnetosphere 231
Manchester coding 27, 280, 284
Marshall and Palmer drop size distribution
225
matched filter 103
matrix condition number 353
matrix rank 353
maximum Doppler frequency 221
maximum entropy method (MEM) 318–319
maximum frequency deviation 80
maximum permissible exposure (MPE) 395
maximum ratio combining (MRC) 164
maximum usable frequency (MUF) 233–234
mechanical tilt 142
media access control (MAC) 366, 407
medium earth orbit satellite 250
mesh network 369
message 4, 17
microstrip antenna 128–130, 249
bandwidth 129
circular polarization 129
microstrip line 8
microwave link 159–162, 181, 193
Miller coding 280, 285
millimeter wave 11
minimum mean square error (MMSE)
equalizer 60
minimum shift keying (MSK) 89–90
misconfiguration attack 375
molded interconnect devices (MID) 147
monopole antenna 119, 146–147, 241, 267
Motley–Keenan indoor attenuation model
213–215
MRC diversity gain 164
multi-band antenna 149
multipath 52, 179–181, 192, 204
multiple access 97–100
multiple beam antennas 144–145, 245,
259–261
multiple input/multiple output (MIMO)
345–361
multiple input single output (MISO) 345
multiple readers 285–288
MUltiple SIgnal Classification (MUSIC)
316–317
multiplexing 94–97
multi-turn loop antenna 124
n
near field communications (NFC) 149, 281
near vertical incidence (NVIS) 234
Nelder–Mead algorithm 339
network interface card (NIC) 366
Nicola Tesla 3–4
noise covariance matrix 303–304
noise eigenvalues 311, 312, 316
noise factor (F) 30
noise figure 30
nonionizing radiation 389
nonrepudiation 376, 380
non return to zero (NRZ) 26, 280
nonsystematic codes 47
null filling 142–144
null-steering 140–142
o
offset quadrature phase shift keying (OQPSK)
87–88
one-time pad encryption 376
one-way encryption 380
on object gain penalty 290
on-off keying (OOK) 26, 79, 279, 284
Open Systems Interconnection (OSI) 407
optimum weights 328–329
429
430
Index
Optimum Working Frequency (OWF) 234
organizationally unique identifier (OUI) 366
orthogonal codes 104
orthogonal frequency division multiple access
(OFDMA) 99–100, 416
orthogonal frequency division multiplexing
(OFDM) 96, 348
OSCAR 1 243
output feedback (OFB) 379
output power minimization algorithm
338–340
over modulation 76
p
packet 21, 367–368
packet radio 97
packet throughput 98–99
parity bit generating matrix 46
parity bits 40–41
parity check matrix 46
Parseval’s theorem 19, 93
partial adaptive nulling 335–337
passband 71
path gain factor 182
path loss 183, 204
path loss exponent 183
penetration depth 392–393
periodogram 304–305
phantom 398–399
phase center 132
phased array 132, 249, 253–255
phase modulation (PM) 84–90
phase modulation index 84
phase shifter 136
𝜋/4 QPSK 89–90
Pisarenko Harmonic Decomposition (PHD)
315–316
plaintext 378–379
planar array 134–135
planar inverted F antenna (PIFA) 122, 147,
398
plane of incidence 175
pointing error 160
point source 116
Poisson process 98–99
polarization loss factor (PLF) 114, 290
polar non return to zero 26
power 28
power loss exponent 214, 216
power minimization 334–340
power spectral density (PSD) 18–20, 92–94,
101
prefix code 23
probability density function (PDF) 207–211
Gaussian 32, 35–36, 207
log-normal 216
Rayleigh 208
Rician 211
processing gain 100
pseudo random noise (PRN) 60, 101, 104,
376, 379, 382
public key cryptography (PKC) 378, 379–380
pulse-interval encoding (PIE) 279
q
quadrature amplitude modulation (QAM)
90–92
quadrature phase shift keying (QPSK)
85–88
quantization error 38
r
radar range equation 292
radiation efficiency 112
radiation resistance 111, 125
Radio Amateur Satellite Corporation
(AMSAT) 243
radio frequency identification (RFID)
history 267–270
passive 267
reader 270, 283, 287–293
tag 270
radio frequency identification (RFID) tag
active 271–272, 278–279
battery-assisted (BAT) 277
chipless 292–295
class 277–278
far field 285–285–295
frequency bands 272
hard 271
HF 284
identifier 274
LF 284
near field 281–285
passive 267, 274–277, 281–285, 288–295
proximity 284
quiet 273
radar cross section 292, 295
read-only (RO) 274
semi-passive 271, 272, 274, 277–278
smart 274
Index
vicinity 284
write-once-read-memory (WORM) 274
radio frequency interference (RFI) 12
rain attenuation 225
rain drop size distribution 225
raised cosine filter 34–56
random search algorithm 335–338
ray tracing 173, 217
reader talk first (RTF) 288
realized gain 112
reconfigurable antenna 340–341
rectangular aperture 125
redundancy checking 42–45
Reed and Solomon code 47
reflector antenna 156–162, 254–255
beam-waveguide 157–158
Cassegrain 157–158
Gregorian 157
offset 158
subreflector 157
refraction 175–176
remote electrical tilt (RET) 142
repetition 40
resolution 304
return to zero (RZ) 26
RF dosimetry 393–396
RF heating 389–393
Rice factor 211
rogue AP (RAP) 375
roll-off factor 54
root MUSIC algorithm 317–318
root raised cosine pulse 56–57
Rotman lens 145, 340
router 367–368, 373
s
sample covariance matrix 332
sample matrix inversion (SMI) 332–334
satellite frequency bands 242
satellite orbits
apogee 246
circular polarization 246
Clarke orbit 247
correction 247
elliptical 246
geostationary orbit 247
geosynchronous orbit 246
inclination 246
low earth orbit (LEO) 246
medium earth orbit (MEO) 246
perigee 246
Scrabble 5
secret key cryptography (SKC) 378–379
sector antenna 142–143, 154–155
selection diversity gain 163
service set identifier (SSID) 374
shadowing 204
shooting and bouncing rays (SBR) 198, 217
signal covariance matrix 303
signal to interference plus noise ratio (SINR)
101
signal to interference ratio (SIR) 33, 326
signal-to-noise and distortion ratio (SINAD)
37
signal to noise ratio (SNR) 29, 101
simplex 95
single input multiple output (SIMO) 345
single input single output (SISO) 345, 347
single sideband (SSB) 78
singular value 353–354, 357, 359
singular value decomposition (SVD) 353
singulation 288
skip distance 230
skywave 227
slot antenna 146
slotted ALOHA 98–99
slow frequency hopping (SFH) 102
smart card 285
smart home 371–372
smart label 270
Snell’s laws 176, 188–189
Software Communications Architecture (SCA)
422
software defined radio (SDR) 419–424
solar flux 230
solar wind 231
source coding 22
SPACEWAY 259–261
spatial multiplexing 346
specific absorption rate (SAR) 393–396,
398–399
spectral efficiency 94
spectrum 8–9
spin stabilization 245, 261–262
spoofing, tampering, repudiation, information
disclosure, DoS, elevation of privilege
(STRIDE) 375
spreading factor 103
Sputnik 241
stamped metal 147–148
standard normal distribution 35
star network 367–368
step size 329
431
432
Index
stream cipher 379
subarray 320
subcarrier 280, 284
subchannel impulse response 347–351
sun fade days 248
sunspots 230
supercapacitor 277
surface roughness 188–189
Surveillance, Epidemiology, and End Results
(SEER) 400
symbol 21
syndrome 46
systematic code 46
system noise temperature 257
t
tag talk first (TTF) 288
taper efficiency 135
television 96
Telstar 243
thermal noise 30, 32, 38, 257
three-axis stabilization 245, 261–262
throughput 98–99
time division duplexing (TDD) 95, 249
time division multiple access (TDMA) 99,
261
time division multiplexing (TDM) 95–96
statistical 97
synchronous 96–97
time-frequency plot 20
total electron content (TEC) 230
Tracking and Data Relay Satellite (TDRS)
245
transmission line 7
transponder 259, 269, 271, 278
transverse electric (TE) 176–177, 182, 184,
189, 200–202
transverse magnetic (TM) 176–177, 182,
184, 189, 200–202
tree network 369
tree walking 288
triple data encryption standard (3DES) 379
tunnel propagation 185–186
turbo code 48
two-shot molding 147–148
type A reference interval (TARI) 279
u
uniform linear array 132
unipolar return to zero 27
universally administered address (UAA)
366
universal product code (UPC) 268–269,
273
uplink (UL) 242, 243, 250, 255, 259, 260,
261
v
variable length code 23
vertical redundancy check (VRC) 42–43
virtual height of the ionospheric layer 228
virtual private network (VPN) 382–383
Visible Human Project (VHP) 398
Viterbi algorithm 48
w
Walsh codes 104
war driving attack 375
waterfilling algorithm 356–360
waveguide 8
weight constraints 335, 337–338
whole body average SAR (WBSAR) 395
wideband antenna 115
Wiener–Hopf solution 329
Wi-Fi 217, 415
Wi-Fi Protected Access (WPA) 382
Wi-Fi Protected Access version 2 (WPA2)
382
wire antenna 117–125
Wired Equivalent Privacy (WEP) 382
wireless local area network (WLAN) 365,
370–374
Wolf number 230
write-once-read-many (WORM) 274
Wullenweber array 305–306
y
Yagi–Uda antenna
120–121
z
zero-forcing equalizer
z-transform 138
59